Mathematics is not simply about memorizing equations. It is about understanding relationships and knowing how to use mathematical tools to solve problems.
A formula can tell you how to calculate the area of a circle, solve a quadratic equation, find the probability of an event, determine the volume of a cylinder, calculate compound interest, or analyze a set of data. But knowing the formula alone is not enough. A student also needs to understand what the formula means, when to use it, how to substitute values correctly, and what mistakes to avoid.
That is the purpose of this guide.
Instead of presenting mathematics formulas as a long list to memorize, this resource explains each important formula using a simple five-part approach:
- Formula — the mathematical relationship.
- Meaning — what the formula and its symbols represent.
- When to use it — the type of problem where the formula is appropriate.
- Worked example — a simple demonstration.
- Common mistake — an error students should avoid.
The guide covers more than 100 formulas from arithmetic, algebra, geometry, coordinate geometry, trigonometry, sequences and series, statistics, probability, financial mathematics, logarithms, exponential functions, and introductory calculus.
Important: Do not try to memorize this entire guide in one sitting. Learn formulas according to the topics you are studying, understand the ideas behind them, and practice using them repeatedly.
Quick Navigation
- Basic Arithmetic Formulas
- Algebra Formulas
- Coordinate Geometry
- Geometry Formulas
- Surface Area and Volume
- Trigonometry
- Sequences and Series
- Statistics
- Probability
- Financial Mathematics
- Logarithms and Exponents
- Basic Calculus
- How to Study Mathematics Formulas
- Common Formula Mistakes
- Frequently Asked Questions
1. Basic Arithmetic Formulas
Arithmetic formulas are among the first mathematical relationships students encounter. They are used in everyday calculations as well as more advanced mathematics.
1.1 Percentage
Formula
[
\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100
]
Meaning
A percentage expresses a quantity as a fraction of 100.
When to use it
Use this formula when you want to determine what percentage one quantity represents of another.
Worked example
A student answers 45 questions correctly out of 50.
[
\text{Percentage}=\frac{45}{50}\times100
]
[
=90%
]
The student scored 90%.
Common mistake
Do not divide by the part instead of the whole.
1.2 Percentage Increase
Formula
[
\text{Percentage Increase}
\frac{\text{New Value}-\text{Original Value}}
{\text{Original Value}}\times100
]
Meaning
This formula tells you how much a quantity has increased relative to its original value.
When to use it
Use it when comparing an old value with a larger new value.
Worked example
A book increases in price from ₦2,000 to ₦2,400.
Increase:
[
2400-2000=400
]
Therefore:
[
\frac{400}{2000}\times100=20%
]
The price increased by 20%.
Common mistake
Use the original value as the denominator, not the new value.
1.3 Percentage Decrease
Formula
[
\text{Percentage Decrease}
\frac{\text{Original Value}-\text{New Value}}
{\text{Original Value}}\times100
]
Meaning
This calculates how much a quantity has decreased relative to its original value.
When to use it
Use it when a price, population, measurement, or other quantity becomes smaller.
Worked example
A school fee decreases from ₦100,000 to ₦90,000.
[
\frac{100000-90000}{100000}\times100=10%
]
The decrease is 10%.
Common mistake
Do not use the new value as the denominator when calculating percentage decrease.
1.4 Average
Formula
[
\text{Average}=\frac{\text{Sum of Values}}{\text{Number of Values}}
]
Meaning
The average, or arithmetic mean, represents the central value obtained by distributing the total equally among all observations.
When to use it
Use it when you need the mean of a collection of numbers.
Worked example
Find the average of 8, 10, 12, 14 and 16.
[
\text{Average}=\frac{8+10+12+14+16}{5}
]
[
=\frac{60}{5}=12
]
Common mistake
Make sure you divide by the correct number of observations.
1.5 Ratio
Formula
[
a:b
]
Meaning
A ratio compares two quantities.
When to use it
Use ratios when comparing quantities of the same type.
Worked example
If a class has 20 boys and 30 girls, the ratio of boys to girls is:
[
20:30=2:3
]
Common mistake
Always simplify a ratio when appropriate, and keep the order of the quantities correct.
1.6 Proportion
Formula
[
\frac{a}{b}=\frac{c}{d}
]
Therefore:
[
ad=bc
]
Meaning
A proportion states that two ratios are equal.
When to use it
Use it to find an unknown quantity when two relationships are proportional.
Worked example
[
\frac{3}{5}=\frac{x}{20}
]
Cross multiply:
[
5x=60
]
[
x=12
]
Common mistake
Cross multiplication must preserve the positions of the numerator and denominator.
1.7 Speed
Formula
[
\text{Speed}=\frac{\text{Distance}}{\text{Time}}
]
Meaning
Speed measures how much distance is covered per unit of time.
When to use it
Use it when distance and time are known and you need to calculate speed.
Worked example
A car travels 240 km in 4 hours.
[
\text{Speed}=\frac{240}{4}=60\text{ km/h}
]
Common mistake
Make sure distance and time use compatible units.
1.8 Distance
Formula
[
\text{Distance}=\text{Speed}\times\text{Time}
]
Worked example
A vehicle travels at 80 km/h for 3 hours.
[
d=80\times3=240\text{ km}
]
Common mistake
Do not multiply speed by time if the units are inconsistent.
1.9 Time
Formula
[
\text{Time}=\frac{\text{Distance}}{\text{Speed}}
]
Worked example
A journey covers 300 km at 75 km/h.
[
t=\frac{300}{75}=4\text{ hours}
]
Common mistake
Check whether the answer should be expressed in hours, minutes, or seconds.
1.10 Simple Interest
Formula
[
I=PRT
]
where:
- (I) = interest
- (P) = principal
- (R) = interest rate expressed as a decimal
- (T) = time
When to use it
Use this formula when interest is calculated only on the original principal.
Worked example
Find the simple interest on ₦50,000 at 10% per year for 2 years.
[
I=50,000(0.10)(2)
]
[
I=₦10,000
]
Common mistake
Convert 10% to 0.10, not 10.
1.11 Amount with Simple Interest
Formula
[
A=P(1+RT)
]
Meaning
The amount is the principal plus the interest earned.
Worked example
Using ₦50,000 at 10% for 2 years:
[
A=50,000(1+0.10\times2)
]
[
A=₦60,000
]
Common mistake
Do not confuse the final amount with the interest alone.
1.12 Profit
Formula
[
\text{Profit}=\text{Selling Price}-\text{Cost Price}
]
Worked example
An item costs ₦8,000 and is sold for ₦10,000.
[
\text{Profit}=10,000-8,000=₦2,000
]
Common mistake
Profit is based on the difference between selling price and cost price.
1.13 Loss
Formula
[
\text{Loss}=\text{Cost Price}-\text{Selling Price}
]
Worked example
An item bought for ₦10,000 is sold for ₦8,500.
[
\text{Loss}=10,000-8,500=₦1,500
]
Common mistake
Do not calculate loss when the selling price is higher than the cost price.
1.14 Profit Percentage
Formula
[
\text{Profit %}
\frac{\text{Profit}}{\text{Cost Price}}\times100
]
Worked example
If profit is ₦2,000 on an item costing ₦8,000:
[
\frac{2000}{8000}\times100=25%
]
Common mistake
Profit percentage is normally calculated using the cost price as the base.
1.15 Loss Percentage
Formula
[
\text{Loss %}
\frac{\text{Loss}}{\text{Cost Price}}\times100
]
Worked example
A ₦10,000 item is sold at a loss of ₦1,500.
[
\frac{1500}{10000}\times100=15%
]
Common mistake
Do not use the selling price as the denominator unless a question specifically asks for a percentage based on selling price.
2. Algebra Formulas
Algebra uses letters and symbols to represent numbers and relationships. It is one of the most important foundations for higher mathematics.
2.1 Distributive Law
Formula
[
a(b+c)=ab+ac
]
Meaning
Multiplication distributes across terms inside brackets.
Worked example
[
3(x+4)=3x+12
]
Common mistake
Students sometimes multiply the first term but forget to multiply the second term.
2.2 Square of a Sum
Formula
[
(a+b)^2=a^2+2ab+b^2
]
Worked example
[
(x+3)^2=x^2+6x+9
]
Common mistake
Do not write (a^2+b^2). The middle term (2ab) is essential.
2.3 Square of a Difference
Formula
[
(a-b)^2=a^2-2ab+b^2
]
Worked example
[
(x-4)^2=x^2-8x+16
]
Common mistake
The final term is positive because a negative number multiplied by itself is positive.
2.4 Difference of Two Squares
Formula
[
a^2-b^2=(a-b)(a+b)
]
Worked example
[
x^2-25=(x-5)(x+5)
]
Common mistake
This formula applies to a difference of squares, not a sum.
2.5 Cube of a Sum
Formula
[
(a+b)^3=a^3+3a^2b+3ab^2+b^3
]
Worked example
[
(x+2)^3=x^3+6x^2+12x+8
]
Common mistake
Do not omit the middle terms.
2.6 Cube of a Difference
Formula
[
(a-b)^3=a^3-3a^2b+3ab^2-b^3
]
Worked example
[
(x-2)^3=x^3-6x^2+12x-8
]
Common mistake
Pay close attention to alternating signs.
2.7 Sum of Two Cubes
Formula
[
a^3+b^3=(a+b)(a^2-ab+b^2)
]
Worked example
[
x^3+8=(x+2)(x^2-2x+4)
]
Common mistake
The signs inside the second bracket are different from those in the first bracket.
2.8 Difference of Two Cubes
Formula
[
a^3-b^3=(a-b)(a^2+ab+b^2)
]
Worked example
[
x^3-27=(x-3)(x^2+3x+9)
]
Common mistake
Do not confuse the signs with the sum-of-cubes identity.
2.9 Product Law of Indices
Formula
[
a^m\times a^n=a^{m+n}
]
Meaning
When multiplying powers with the same base, add the exponents.
Worked example
[
x^3\times x^4=x^7
]
Common mistake
Do not multiply the exponents.
2.10 Quotient Law of Indices
Formula
[
\frac{a^m}{a^n}=a^{m-n}
]
Worked example
[
\frac{x^7}{x^3}=x^4
]
Common mistake
Subtract the exponents in the correct order.
2.11 Power of a Power
Formula
[
(a^m)^n=a^{mn}
]
Worked example
[
(x^2)^3=x^6
]
Common mistake
Multiply the exponents rather than adding them.
2.12 Zero Exponent
Formula
[
a^0=1
]
for (a\neq0).
Worked example
[
7^0=1
]
Common mistake
Do not assume (0^0=1) in ordinary algebra; that expression requires separate context.
2.13 Negative Exponent
Formula
[
a^{-n}=\frac{1}{a^n}
]
Worked example
[
x^{-3}=\frac1{x^3}
]
Common mistake
A negative exponent does not make the answer negative. It indicates a reciprocal.
2.14 Fractional Exponent
Formula
[
a^{1/n}=\sqrt[n]{a}
]
Worked example
[
16^{1/2}=\sqrt{16}=4
]
Common mistake
The denominator of the fractional exponent determines the root.
2.15 General Fractional Exponent
Formula
[
a^{m/n}=\sqrt[n]{a^m}
]
Worked example
[
8^{2/3}=\sqrt[3]{8^2}=4
]
Common mistake
Do not confuse the numerator and denominator of the exponent.
2.16 Quadratic Formula
Formula
For:
[
ax^2+bx+c=0
]
[
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
]
Meaning
The quadratic formula provides the solutions of a quadratic equation.
When to use it
Use it when a quadratic equation cannot easily be solved by factorization or when a general method is required.
Worked example
Solve:
[
x^2-5x+6=0
]
Here:
[
a=1,\quad b=-5,\quad c=6
]
[
x=\frac{5\pm\sqrt{25-24}}2
]
[
x=\frac{5\pm1}{2}
]
Therefore:
[
x=3\quad\text{or}\quad x=2
]
Common mistake
A common error is forgetting that (-b) changes the sign of (b).
2.17 Discriminant
Formula
[
D=b^2-4ac
]
Meaning
The discriminant indicates the nature of the roots of a quadratic equation.
- (D>0): two distinct real roots.
- (D=0): one repeated real root.
- (D<0): no real roots.
Worked example
For:
[
x^2-4x+4=0
]
[
D=(-4)^2-4(1)(4)=0
]
Therefore, there is one repeated real root.
Common mistake
Remember that (b) must be squared before subtracting (4ac).
3. Coordinate Geometry Formulas
Coordinate geometry combines algebra and geometry to describe points, lines, distances, and shapes on a coordinate plane.
3.1 Distance Between Two Points
Formula
[
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
]
Meaning
This calculates the straight-line distance between two points.
Worked example
For ((1,2)) and ((4,6)):
[
d=\sqrt{(4-1)^2+(6-2)^2}
]
[
=\sqrt{9+16}
]
[
=5
]
Common mistake
Do not forget the square root at the end.
3.2 Midpoint Formula
Formula
[
M=
\left(
\frac{x_1+x_2}{2},
\frac{y_1+y_2}{2}
\right)
]
Meaning
The midpoint is the point exactly halfway between two coordinates.
Worked example
For ((2,4)) and ((8,10)):
[
M=\left(\frac{2+8}{2},\frac{4+10}{2}\right)
]
[
M=(5,7)
]
Common mistake
Calculate the (x)-coordinates and (y)-coordinates separately.
3.3 Gradient or Slope
Formula
[
m=\frac{y_2-y_1}{x_2-x_1}
]
Meaning
The gradient measures how steeply a line rises or falls.
Worked example
For ((1,2)) and ((5,10)):
[
m=\frac{10-2}{5-1}=2
]
Common mistake
Maintain the same order when subtracting the (x) and (y) coordinates.
3.4 Slope-Intercept Form
Formula
[
y=mx+c
]
Meaning
Here (m) represents the slope and (c) represents the (y)-intercept.
Worked example
If:
[
m=3,\quad c=2
]
then:
[
y=3x+2
]
Common mistake
Do not confuse the slope (m) with the (y)-intercept (c).
3.5 Point-Slope Form
Formula
[
y-y_1=m(x-x_1)
]
When to use it
Use it when you know one point on a line and its slope.
Worked example
For slope (2) passing through ((3,4)):
[
y-4=2(x-3)
]
Common mistake
The signs inside the brackets are easy to mishandle.
3.6 Equation of a Circle
Formula
[
(x-h)^2+(y-k)^2=r^2
]
Meaning
The center is ((h,k)), and (r) is the radius.
Worked example
A circle with center ((2,3)) and radius 5 has:
[
(x-2)^2+(y-3)^2=25
]
Common mistake
Remember that the signs in the equation are opposite to the signs of the center coordinates.
4. Geometry Formulas
Geometry deals with shapes, angles, lengths, areas, and spatial relationships.
4.1 Perimeter of a Rectangle
Formula
[
P=2(l+w)
]
Meaning
The perimeter is the total distance around the rectangle.
Worked example
For (l=8) cm and (w=5) cm:
[
P=2(8+5)=26\text{ cm}
]
Common mistake
Perimeter uses linear units, not square units.
4.2 Area of a Rectangle
Formula
[
A=lw
]
Worked example
For a rectangle measuring 8 cm by 5 cm:
[
A=8\times5=40\text{ cm}^2
]
Common mistake
Area is measured in square units.
4.3 Perimeter of a Square
Formula
[
P=4a
]
Worked example
A square with side 6 cm:
[
P=4(6)=24\text{ cm}
]
Common mistake
Do not use (a^2); that calculates area.
4.4 Area of a Square
Formula
[
A=a^2
]
Worked example
For a side of 6 cm:
[
A=6^2=36\text{ cm}^2
]
Common mistake
Do not multiply by 4. That is the perimeter formula.
4.5 Area of a Triangle
Formula
[
A=\frac12bh
]
Meaning
(b) is the base and (h) is the perpendicular height.
Worked example
For (b=10) cm and (h=6) cm:
[
A=\frac12(10)(6)=30\text{ cm}^2
]
Common mistake
The height must be perpendicular to the chosen base.
4.6 Area of a Parallelogram
Formula
[
A=bh
]
Worked example
For (b=12) cm and (h=5) cm:
[
A=60\text{ cm}^2
]
Common mistake
Use perpendicular height, not the slanted side.
4.7 Area of a Trapezium/Trapezoid
Formula
[
A=\frac12(a+b)h
]
Meaning
(a) and (b) are the parallel sides.
Worked example
If the parallel sides are 8 cm and 12 cm and the height is 5 cm:
[
A=\frac12(8+12)(5)=50\text{ cm}^2
]
Common mistake
Only the parallel sides are added.
4.8 Area of a Rhombus
Formula
[
A=\frac12d_1d_2
]
Worked example
If the diagonals are 10 cm and 8 cm:
[
A=\frac12(10)(8)=40\text{ cm}^2
]
Common mistake
Do not confuse diagonals with side lengths.
4.9 Circumference of a Circle
Formula
[
C=2\pi r
]
Meaning
Circumference is the distance around a circle.
Worked example
For (r=7) cm:
[
C=14\pi\approx43.98\text{ cm}
]
Common mistake
Do not use (r^2); that belongs to the area formula.
4.10 Area of a Circle
Formula
[
A=\pi r^2
]
Worked example
For (r=7) cm:
[
A=49\pi\approx153.94\text{ cm}^2
]
Common mistake
Do not confuse circumference with area.
4.11 Diameter of a Circle
Formula
[
d=2r
]
Worked example
If the radius is 9 cm:
[
d=18\text{ cm}
]
Common mistake
Do not divide the radius by 2.
4.12 Arc Length
Formula
For an angle measured in degrees:
[
L=\frac{\theta}{360^\circ}(2\pi r)
]
Meaning
Arc length is the distance along part of a circle.
Worked example
For (r=10) cm and (\theta=90^\circ):
[
L=\frac{90}{360}(20\pi)
]
[
L=5\pi\approx15.71\text{ cm}
]
Common mistake
Make sure the angle is in degrees when using the (360^\circ) version.
4.13 Sector Area
Formula
[
A=\frac{\theta}{360^\circ}\pi r^2
]
Worked example
For (r=6) cm and (\theta=60^\circ):
[
A=\frac{60}{360}\pi(36)
]
[
A=6\pi\approx18.85\text{ cm}^2
]
Common mistake
Do not forget to divide the angle by (360^\circ).
4.14 Pythagorean Theorem
Formula
[
a^2+b^2=c^2
]
Meaning
For a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
When to use it
Use it only for right-angled triangles.
Worked example
If the shorter sides are 3 cm and 4 cm:
[
c^2=3^2+4^2=25
]
[
c=5\text{ cm}
]
Common mistake
The hypotenuse must be opposite the right angle.
4.15 Sum of Interior Angles of a Triangle
Formula
[
A+B+C=180^\circ
]
Worked example
If two angles are (60^\circ) and (70^\circ):
[
C=180-60-70=50^\circ
]
Common mistake
A triangle always has three interior angles whose sum is (180^\circ).
4.16 Sum of Interior Angles of a Polygon
Formula
[
S=(n-2)180^\circ
]
Worked example
For a hexagon:
[
S=(6-2)180=720^\circ
]
Common mistake
Use the number of sides for (n), not the number of angles you have already calculated.
4.17 Interior Angle of a Regular Polygon
Formula
[
\frac{(n-2)180^\circ}{n}
]
Worked example
For a regular pentagon:
[
\frac{(5-2)180}{5}=108^\circ
]
Common mistake
This formula applies when all interior angles are equal.
4.18 Exterior Angle of a Regular Polygon
Formula
[
\frac{360^\circ}{n}
]
Worked example
For a regular hexagon:
[
\frac{360}{6}=60^\circ
]
Common mistake
Do not use the interior-angle formula.
5. Surface Area and Volume Formulas
Surface area measures the outside surface of a three-dimensional object, while volume measures the amount of space it occupies.
5.1 Volume of a Cube
Formula
[
V=a^3
]
Worked example
For a cube with side 4 cm:
[
V=4^3=64\text{ cm}^3
]
Common mistake
Volume uses cubic units.
5.2 Surface Area of a Cube
Formula
[
SA=6a^2
]
Worked example
For (a=4) cm:
[
SA=6(16)=96\text{ cm}^2
]
Common mistake
Do not use (a^3), which calculates volume.
5.3 Volume of a Cuboid
Formula
[
V=lwh
]
Worked example
For (l=10), (w=4), and (h=3):
[
V=10(4)(3)=120\text{ cm}^3
]
Common mistake
Check that all three dimensions are expressed in the same units.
5.4 Surface Area of a Cuboid
Formula
[
SA=2(lw+lh+wh)
]
Worked example
For (l=5), (w=3), (h=2):
[
SA=2(15+10+6)=62\text{ cm}^2
]
Common mistake
Remember to multiply the sum by 2.
5.5 Volume of a Cylinder
Formula
[
V=\pi r^2h
]
Worked example
For (r=3) cm and (h=10) cm:
[
V=\pi(3^2)(10)=90\pi
]
[
V\approx282.74\text{ cm}^3
]
Common mistake
Do not use the diameter as the radius.
5.6 Curved Surface Area of a Cylinder
Formula
[
CSA=2\pi rh
]
Worked example
For (r=3) cm and (h=10) cm:
[
CSA=60\pi\approx188.50\text{ cm}^2
]
Common mistake
Curved surface area does not include the two circular ends.
5.7 Total Surface Area of a Cylinder
Formula
[
TSA=2\pi r(h+r)
]
Worked example
For (r=3), (h=10):
[
TSA=2\pi(3)(13)=78\pi
]
Common mistake
Do not confuse total surface area with curved surface area.
5.8 Volume of a Cone
Formula
[
V=\frac13\pi r^2h
]
Worked example
For (r=3) cm and (h=8) cm:
[
V=\frac13\pi(9)(8)=24\pi
]
Common mistake
The factor (\frac13) is essential.
5.9 Slant Height of a Cone
Formula
[
l=\sqrt{r^2+h^2}
]
When to use it
Use it when the radius and vertical height are known and the slant height is required.
Worked example
For (r=3) cm and (h=4) cm:
[
l=\sqrt{9+16}=5\text{ cm}
]
Common mistake
Do not confuse vertical height with slant height.
5.10 Curved Surface Area of a Cone
Formula
[
CSA=\pi rl
]
Worked example
For (r=3) cm and (l=5) cm:
[
CSA=15\pi\text{ cm}^2
]
Common mistake
You need the slant height, not the vertical height.
5.11 Total Surface Area of a Cone
Formula
[
TSA=\pi r(l+r)
]
Worked example
For (r=3) and (l=5):
[
TSA=\pi(3)(8)=24\pi\text{ cm}^2
]
Common mistake
The circular base must be included for total surface area.
5.12 Volume of a Sphere
Formula
[
V=\frac43\pi r^3
]
Worked example
For (r=3) cm:
[
V=\frac43\pi(27)=36\pi\text{ cm}^3
]
Common mistake
Use (r^3), not (r^2).
5.13 Surface Area of a Sphere
Formula
[
SA=4\pi r^2
]
Worked example
For (r=5) cm:
[
SA=4\pi(25)=100\pi\text{ cm}^2
]
Common mistake
Do not use the volume formula.
5.14 Volume of a Hemisphere
Formula
[
V=\frac23\pi r^3
]
Worked example
For (r=3) cm:
[
V=\frac23\pi(27)=18\pi\text{ cm}^3
]
Common mistake
A hemisphere has half the volume of a sphere of the same radius.
5.15 Total Surface Area of a Hemisphere
Formula
[
TSA=3\pi r^2
]
Meaning
This includes both the curved surface and circular base.
Worked example
For (r=4) cm:
[
TSA=3\pi(16)=48\pi\text{ cm}^2
]
Common mistake
The curved surface alone is (2\pi r^2); total surface area includes the base.
6. Trigonometry Formulas
Trigonometry studies relationships between angles and sides of triangles. It is important in mathematics, physics, engineering, surveying, construction, and many other fields.
6.1 Sine Ratio
Formula
[
\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}
]
Meaning
For a right-angled triangle, sine relates the opposite side to the hypotenuse.
Worked example
If the opposite side is 6 cm and the hypotenuse is 10 cm:
[
\sin\theta=\frac6{10}=0.6
]
Common mistake
The hypotenuse is always opposite the right angle.
6.2 Cosine Ratio
Formula
[
\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}
]
Worked example
If adjacent = 8 and hypotenuse = 10:
[
\cos\theta=\frac8{10}=0.8
]
Common mistake
Make sure you identify the adjacent side relative to the angle being considered.
6.3 Tangent Ratio
Formula
[
\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}
]
Worked example
If opposite = 3 and adjacent = 4:
[
\tan\theta=\frac34
]
Common mistake
Do not include the hypotenuse in the tangent ratio.
6.4 Pythagorean Trigonometric Identity
Formula
[
\sin^2\theta+\cos^2\theta=1
]
Worked example
If:
[
\sin\theta=\frac35
]
then:
[
\cos^2\theta=1-\frac9{25}=\frac{16}{25}
]
so:
[
\cos\theta=\frac45
]
for an acute angle.
Common mistake
(\sin^2\theta) means ((\sin\theta)^2), not (\sin(\theta^2)).
6.5 Tangent Identity
Formula
[
\tan\theta=\frac{\sin\theta}{\cos\theta}
]
Worked example
If (\sin\theta=0.6) and (\cos\theta=0.8):
[
\tan\theta=\frac{0.6}{0.8}=0.75
]
Common mistake
The identity requires (\cos\theta\neq0).
6.6 Secant Identity
Formula
[
\sec\theta=\frac1{\cos\theta}
]
Worked example
If:
[
\cos\theta=\frac12
]
then:
[
\sec\theta=2
]
Common mistake
Do not confuse secant with sine or cosecant.
6.7 Cosecant Identity
Formula
[
\csc\theta=\frac1{\sin\theta}
]
Worked example
If:
[
\sin\theta=\frac45
]
then:
[
\csc\theta=\frac54
]
Common mistake
Cosecant is the reciprocal of sine.
6.8 Cotangent Identity
Formula
[
\cot\theta=\frac1{\tan\theta}
]
Worked example
If:
[
\tan\theta=2
]
then:
[
\cot\theta=\frac12
]
Common mistake
Do not confuse cotangent with cosine.
6.9 Sine Rule
Formula
[
\frac{a}{\sin A}
\frac{b}{\sin B}
\frac{c}{\sin C}
]
When to use it
Use it when you know suitable opposite side-angle pairs in a non-right triangle.
Worked example
If:
[
\frac{a}{\sin30^\circ}=\frac{10}{\sin50^\circ}
]
then:
[
a=\frac{10\sin30^\circ}{\sin50^\circ}
]
Common mistake
Side (a) must correspond to angle (A), side (b) to (B), and side (c) to (C).
6.10 Cosine Rule
Formula
[
c^2=a^2+b^2-2ab\cos C
]
When to use it
It is especially useful when two sides and the included angle are known, or when all three sides are known and an angle is required.
Worked example
Suppose:
[
a=5,\quad b=7,\quad C=60^\circ
]
Then:
[
c^2=25+49-2(5)(7)\cos60^\circ
]
[
c^2=74-35=39
]
[
c=\sqrt{39}
]
Common mistake
The angle (C) must be the angle between sides (a) and (b).
6.11 Area of a Triangle Using Two Sides and Included Angle
Formula
[
A=\frac12ab\sin C
]
Worked example
If (a=8), (b=10), and (C=30^\circ):
[
A=\frac12(8)(10)(0.5)=20
]
Common mistake
The angle must be between the two given sides.
6.12 Double-Angle Formula for Sine
Formula
[
\sin2\theta=2\sin\theta\cos\theta
]
Worked example
If (\sin\theta=\frac35) and (\cos\theta=\frac45):
[
\sin2\theta=2\left(\frac35\right)\left(\frac45\right)
=\frac{24}{25}
]
Common mistake
Do not write (\sin2\theta=2\sin\theta).
6.13 Double-Angle Formula for Cosine
Formula
[
\cos2\theta=\cos^2\theta-\sin^2\theta
]
Equivalent forms include:
[
\cos2\theta=2\cos^2\theta-1
]
and:
[
\cos2\theta=1-2\sin^2\theta
]
Worked example
If:
[
\cos\theta=\frac45,\quad \sin\theta=\frac35
]
then:
[
\cos2\theta=\frac{16}{25}-\frac9{25}
=\frac7{25}
]
Common mistake
Do not confuse the double-angle formula with the square of the cosine.
7. Sequences and Series
Sequences are ordered lists of numbers. A series is obtained by adding terms of a sequence.
7.1 Arithmetic Sequence: nth Term
Formula
[
a_n=a+(n-1)d
]
Meaning
- (a_n) = nth term
- (a) = first term
- (d) = common difference
- (n) = term number
Worked example
For:
[
3,7,11,15,\ldots
]
(a=3), (d=4).
The 10th term is:
[
a_{10}=3+(10-1)(4)=39
]
Common mistake
Use (n-1), not (n).
7.2 Sum of an Arithmetic Sequence
Formula
[
S_n=\frac n2[2a+(n-1)d]
]
Worked example
Find the sum of the first 10 terms of:
[
3,7,11,\ldots
]
[
S_{10}=\frac{10}{2}[6+36]
]
[
=210
]
Common mistake
Do not confuse the sum formula with the nth-term formula.
7.3 Arithmetic Series Using the Last Term
Formula
[
S_n=\frac n2(a+l)
]
Meaning
(l) is the last term.
Worked example
If (a=3), (l=39), (n=10):
[
S_{10}=\frac{10}{2}(3+39)=210
]
Common mistake
Make sure (l) is actually the final term being included.
7.4 Geometric Sequence: nth Term
Formula
[
a_n=ar^{n-1}
]
Meaning
- (a) = first term
- (r) = common ratio
Worked example
For:
[
2,6,18,54,\ldots
]
(a=2), (r=3).
The fifth term is:
[
a_5=2(3)^4=162
]
Common mistake
A geometric sequence uses multiplication by a common ratio, not addition by a common difference.
7.5 Finite Geometric Series
Formula
[
S_n=a\frac{1-r^n}{1-r}
]
for (r\neq1).
Worked example
For:
[
2+6+18+54
]
[
S_4=2\frac{1-3^4}{1-3}
]
[
=80
]
Common mistake
Do not use the arithmetic-series formula for a geometric sequence.
7.6 Infinite Geometric Series
Formula
[
S_\infty=\frac{a}{1-r}
]
provided:
[
|r|<1
]
Worked example
For:
[
10+5+2.5+\cdots
]
[
S_\infty=\frac{10}{1-0.5}=20
]
Common mistake
An infinite geometric series has a finite sum only when (|r|<1).
7.7 Sum of the First (n) Positive Integers
Formula
[
S=\frac{n(n+1)}2
]
Worked example
The sum from 1 to 10 is:
[
\frac{10(11)}2=55
]
Common mistake
This formula starts with 1 and applies to consecutive positive integers.
7.8 Sum of the First (n) Squares
Formula
[
S=\frac{n(n+1)(2n+1)}6
]
Worked example
For the first five squares:
[
S=\frac{5(6)(11)}6=55
]
Indeed:
[
1+4+9+16+25=55
]
Common mistake
Do not confuse this with the sum of the first (n) integers.
7.9 Sum of the First (n) Cubes
Formula
[
S=\left[\frac{n(n+1)}2\right]^2
]
Worked example
For the first four cubes:
[
S=\left[\frac{4(5)}2\right]^2
=10^2=100
]
Common mistake
The entire fraction must be squared.
8. Statistics Formulas
Statistics provides methods for collecting, describing, and interpreting numerical data.
8.1 Mean
Formula
[
\bar{x}=\frac{\sum x}{n}
]
Meaning
The mean is the sum of all observations divided by the number of observations.
Worked example
For 4, 6, 8, 10:
[
\bar{x}=\frac{28}{4}=7
]
Common mistake
Count the observations carefully.
8.2 Weighted Mean
Formula
[
\bar{x}_w=\frac{\sum wx}{\sum w}
]
When to use it
Use it when some observations contribute more heavily than others.
Worked example
Suppose a test score of 80 has weight 2 and a project score of 70 has weight 1:
[
\bar{x}_w=\frac{2(80)+1(70)}{2+1}
]
[
=\frac{230}{3}\approx76.67
]
Common mistake
Do not divide by the number of observations; divide by the total weight.
8.3 Population Variance
Formula
[
\sigma^2=\frac{\sum(x-\mu)^2}{N}
]
Meaning
Variance measures the average squared distance of values from the population mean.
When to use it
Use it when the data represents the entire population being studied.
Worked example
For a small population, calculate each deviation from the mean, square the deviations, add them, and divide by (N).
Common mistake
Do not use (n-1) for a population variance.
8.4 Population Standard Deviation
Formula
[
\sigma=\sqrt{\frac{\sum(x-\mu)^2}{N}}
]
Meaning
Standard deviation is the square root of variance and expresses spread in the same units as the original data.
Worked example
For data (2,4,6):
Mean:
[
\mu=4
]
Variance:
[
\frac{(2-4)^2+(4-4)^2+(6-4)^2}{3}
=\frac83
]
Therefore:
[
\sigma=\sqrt{\frac83}\approx1.63
]
Common mistake
Do not forget to take the square root when moving from variance to standard deviation.
8.5 Sample Variance
Formula
[
s^2=
\frac{\sum(x-\bar{x})^2}{n-1}
]
Meaning
Sample variance estimates population variability from a sample.
When to use it
Use it when your data represents a sample rather than the entire population.
Common mistake
The denominator is (n-1), not (n).
8.6 Sample Standard Deviation
Formula
[
s=
\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}
]
Worked example
For sample data (2,4,6), the mean is 4:
[
s=\sqrt{\frac{8}{2}}=2
]
Common mistake
Do not use the population denominator when calculating sample standard deviation.
8.7 Range
Formula
[
\text{Range}=\text{Maximum}-\text{Minimum}
]
Worked example
For:
[
5,8,12,15,20
]
[
\text{Range}=20-5=15
]
Common mistake
The range is not the average of the maximum and minimum.
8.8 Z-Score
Formula
[
z=\frac{x-\mu}{\sigma}
]
Meaning
A z-score tells you how many standard deviations a value is above or below the population mean.
Worked example
If (x=70), (\mu=60), and (\sigma=5):
[
z=\frac{70-60}{5}=2
]
The value is 2 standard deviations above the mean.
Common mistake
Keep the sign. A negative z-score indicates a value below the mean.
8.9 Coefficient of Variation
Formula
[
CV=\frac{\sigma}{\mu}\times100%
]
Meaning
The coefficient of variation measures standard deviation relative to the mean.
Worked example
If mean = 50 and standard deviation = 5:
[
CV=\frac5{50}\times100=10%
]
Common mistake
Do not forget to multiply by 100 when expressing CV as a percentage.
9. Probability Formulas
Probability measures the likelihood of events occurring.
9.1 Basic Probability
Formula
[
P(A)=\frac{\text{Favorable Outcomes}}
{\text{Total Possible Outcomes}}
]
when outcomes are equally likely.
Worked example
The probability of rolling a 4 on a fair six-sided die is:
[
P(4)=\frac16
]
Common mistake
The numerator counts favorable outcomes, not all outcomes.
9.2 Complement Rule
Formula
[
P(A’)=1-P(A)
]
Meaning
The complement is the probability that an event does not occur.
Worked example
If the probability of rain is 0.3:
[
P(\text{no rain})=1-0.3=0.7
]
Common mistake
Probabilities must remain between 0 and 1.
9.3 Addition Rule
Formula
[
P(A\cup B)=P(A)+P(B)-P(A\cap B)
]
Meaning
This calculates the probability that at least one of two events occurs.
Worked example
If:
[
P(A)=0.5,\quad P(B)=0.4,\quad P(A\cap B)=0.2
]
then:
[
P(A\cup B)=0.5+0.4-0.2=0.7
]
Common mistake
Do not add the probabilities without subtracting the overlap when events can occur together.
9.4 Probability of Independent Events
Formula
[
P(A\cap B)=P(A)P(B)
]
Meaning
For independent events, the occurrence of one does not change the probability of the other.
Worked example
The probability of getting heads twice with a fair coin is:
[
\frac12\times\frac12=\frac14
]
Common mistake
Only use multiplication this way when independence applies.
9.5 Conditional Probability
Formula
[
P(A|B)=\frac{P(A\cap B)}{P(B)}
]
Meaning
This is the probability of (A) occurring given that (B) has occurred.
Worked example
If:
[
P(A\cap B)=0.2
]
and:
[
P(B)=0.5
]
then:
[
P(A|B)=\frac{0.2}{0.5}=0.4
]
Common mistake
The denominator is (P(B)), because the condition is “given (B).”
9.6 Permutations
Formula
[
{}_nP_r=\frac{n!}{(n-r)!}
]
Meaning
Permutations count arrangements where order matters.
Worked example
How many ways can 3 students be selected and arranged from 5?
[
{}_5P_3=\frac{5!}{2!}=60
]
Common mistake
Use permutations when order matters.
9.7 Combinations
Formula
[
{}_nC_r=\frac{n!}{r!(n-r)!}
]
Meaning
Combinations count selections where order does not matter.
Worked example
Choose 3 students from 5:
[
{}_5C_3=\frac{5!}{3!2!}=10
]
Common mistake
Do not use permutations when the order of selection is irrelevant.
9.8 Factorial
Formula
[
n!=n(n-1)(n-2)\cdots2(1)
]
and:
[
0!=1
]
Worked example
[
5!=5\times4\times3\times2\times1=120
]
Common mistake
Factorial is multiplication, not exponentiation.
10. Financial Mathematics Formulas
Financial mathematics helps students understand savings, loans, investment, interest, profit, and other real-world financial calculations.
10.1 Compound Interest
Formula
[
A=P\left(1+\frac rn\right)^{nt}
]
Meaning
- (A) = final amount
- (P) = principal
- (r) = annual interest rate as a decimal
- (n) = number of compounding periods per year
- (t) = time in years
When to use it
Use it when interest is added to the account periodically and future interest is calculated on the accumulated amount.
Worked example
Suppose ₦100,000 is invested at 10% annually for 2 years with annual compounding:
[
A=100000(1.10)^2
]
[
A=₦121,000
]
Common mistake
Do not use 10 instead of 0.10 for the interest rate.
10.2 Compound Interest Earned
Formula
[
I=A-P
]
Worked example
If the final amount is ₦121,000 and the original investment was ₦100,000:
[
I=121000-100000=₦21,000
]
Common mistake
The amount (A) includes the original principal.
10.3 Percentage Change
Formula
[
\text{Percentage Change}
\frac{\text{New Value}-\text{Original Value}}
{\text{Original Value}}\times100
]
Worked example
A quantity changes from 200 to 250:
[
\frac{250-200}{200}\times100=25%
]
Common mistake
Always identify which value is the original value.
11. Logarithms and Exponential Formulas
Logarithms are particularly important in algebra, science, economics, statistics, and advanced mathematics.
11.1 Product Rule for Logarithms
Formula
[
\log_b(xy)=\log_bx+\log_by
]
Meaning
The logarithm of a product can be written as the sum of the logarithms.
Worked example
[
\log_{10}(1000)=\log_{10}(10\times100)
]
[
=1+2=3
]
Common mistake
The product rule uses addition, not multiplication.
11.2 Quotient Rule for Logarithms
Formula
[
\log_b\left(\frac{x}{y}\right)
\log_bx-\log_by
]
Worked example
[
\log_{10}(100)=\log_{10}(1000)-\log_{10}(10)
]
[
=3-1=2
]
Common mistake
Division inside the logarithm becomes subtraction outside it.
11.3 Power Rule for Logarithms
Formula
[
\log_b(x^n)=n\log_bx
]
Worked example
[
\log_{10}(100^2)
=2\log_{10}(100)
=4
]
Common mistake
The exponent moves in front as a multiplier.
11.4 Change of Base Formula
Formula
[
\log_bx=\frac{\log_ax}{\log_ab}
]
When to use it
Use it when your calculator does not have a button for the required base.
Worked example
[
\log_2 8=
\frac{\log 8}{\log2}=3
]
Common mistake
The denominator must be (\log_a b), not (\log_b a).
11.5 Definition of a Logarithm
Formula
[
\log_bx=y
\iff b^y=x
]
Worked example
[
\log_2 16=4
]
because:
[
2^4=16
]
Common mistake
The base of the logarithm must be positive and not equal to 1 in the usual real-number definition.
11.6 Exponential Growth
Formula
[
A=A_0e^{kt}
]
Meaning
This models continuous exponential growth.
Worked example
If (A_0=100), (k=0.05), and (t=10):
[
A=100e^{0.5}
]
which is approximately:
[
A\approx164.87
]
Common mistake
Make sure (k) and (t) use compatible time units.
11.7 Exponential Decay
Formula
[
A=A_0e^{-kt}
]
Meaning
This models continuous exponential decrease.
Worked example
If (A_0=100), (k=0.05), and (t=10):
[
A=100e^{-0.5}\approx60.65
]
Common mistake
The exponent is negative for decay.
12. Basic Calculus Formulas
Calculus studies change and accumulation. These formulas are useful for students progressing to advanced mathematics and university-level subjects.
12.1 Derivative of a Constant
Formula
[
\frac{d}{dx}(c)=0
]
Meaning
A constant does not change as (x) changes.
Worked example
[
\frac{d}{dx}(7)=0
]
Common mistake
Do not confuse a constant with a variable term.
12.2 Power Rule
Formula
[
\frac{d}{dx}(x^n)=nx^{n-1}
]
Meaning
The power is multiplied by the coefficient, and the exponent is reduced by one.
Worked example
[
\frac{d}{dx}(x^4)=4x^3
]
Common mistake
Do not forget to reduce the exponent by 1.
12.3 Derivative of a Sum
Formula
[
\frac{d}{dx}[f(x)+g(x)]
f'(x)+g'(x)
]
Worked example
If:
[
f(x)=x^2+3x
]
then:
[
f'(x)=2x+3
]
Common mistake
Differentiate each term separately.
12.4 Product Rule
Formula
[
\frac{d}{dx}[f(x)g(x)]
f'(x)g(x)+f(x)g'(x)
]
When to use it
Use it when two functions are multiplied together.
Worked example
Let:
[
f(x)=x^2,\quad g(x)=x^3
]
Then:
[
\frac{d}{dx}(x^2x^3)
2x(x^3)+x^2(3x^2)
]
[
=5x^4
]
Common mistake
Do not simply differentiate each factor and multiply the derivatives.
12.5 Quotient Rule
Formula
[
\frac{d}{dx}
\left[\frac{f(x)}{g(x)}\right]
\frac{f'(x)g(x)-f(x)g'(x)}
{[g(x)]^2}
]
When to use it
Use it when one function is divided by another.
Common mistake
The order in the numerator matters.
12.6 Derivative of (e^x)
Formula
[
\frac{d}{dx}(e^x)=e^x
]
Worked example
[
\frac{d}{dx}(5e^x)=5e^x
]
Common mistake
A constant multiplier remains in the derivative.
12.7 Derivative of (\ln x)
Formula
[
\frac{d}{dx}(\ln x)=\frac1x
]
Worked example
[
\frac{d}{dx}(\ln x^2)=\frac{2}{x}
]
for (x>0), when using the logarithm simplification in this form.
Common mistake
Remember that the derivative of (\ln x) is (1/x), not (\ln(x-1)).
12.8 Basic Integration Rule
Formula
[
\int x^n,dx=
\frac{x^{n+1}}{n+1}+C
]
for:
[
n\neq-1
]
Meaning
Integration can be viewed as the reverse process of differentiation.
Worked example
[
\int x^2dx=\frac{x^3}{3}+C
]
Common mistake
Always include the constant (C) for an indefinite integral.
12.9 Integral of (1/x)
Formula
[
\int\frac1x,dx=\ln|x|+C
]
Worked example
[
\int\frac1x,dx=\ln|x|+C
]
Common mistake
The ordinary power-rule formula cannot be used when (n=-1).
12.10 Fundamental Idea of Definite Integration
Formula
[
\int_a^b f(x),dx=F(b)-F(a)
]
where:
[
F'(x)=f(x)
]
Meaning
A definite integral can calculate accumulated quantity, such as area under a curve, when the appropriate conditions are satisfied.
Worked example
[
\int_0^2 x,dx
]
An antiderivative is:
[
F(x)=\frac{x^2}{2}
]
Therefore:
[
F(2)-F(0)=2
]
Common mistake
Do not forget to evaluate the antiderivative at both limits.
13. How to Study and Remember Mathematics Formulas
A formula sheet is useful, but simply reading formulas repeatedly is not the most effective way to learn mathematics.
13.1 Understand the Formula First
Ask yourself:
- What does the formula calculate?
- What does each symbol mean?
- What information does the question give me?
- What information am I being asked to find?
- What units should the answer have?
Understanding the formula makes it easier to remember and apply.
13.2 Learn Formulas by Topic
Instead of trying to memorize 100 formulas together, divide them into groups.
For example:
Day 1: Arithmetic
Day 2: Algebra
Day 3: Geometry
Day 4: Trigonometry
Day 5: Statistics
Day 6: Probability
Day 7: Revision and practice
Adapt the schedule to your own course and examination timetable.
13.3 Use Active Recall
Close your formula sheet and try to write the formula from memory.
Then check your answer.
This is more effective than simply looking at the formula repeatedly.
13.4 Practice Formula Substitution
After learning a formula, immediately solve a simple problem.
For example, after learning:
[
A=\pi r^2
]
try calculating the area of circles with radii of 3 cm, 5 cm, and 10 cm.
13.5 Learn the Units
Units can help identify mistakes.
Examples:
- Length → cm, m, km
- Area → cm², m², km²
- Volume → cm³, m³
- Speed → km/h, m/s
- Time → seconds, minutes, hours
If your answer to an area question is expressed in centimetres rather than square centimetres, check your work.
13.6 Practice Without Immediately Looking at the Answer
Try to solve the problem yourself first.
When you make a mistake, don’t simply erase it. Identify why you made the mistake.
Was it:
- the wrong formula?
- incorrect substitution?
- arithmetic?
- units?
- calculator mode?
- algebraic manipulation?
Understanding your errors is an important part of mathematical learning.
14. Common Mathematics Formula Mistakes
14.1 Memorizing Without Understanding
A student may remember:
[
A=\pi r^2
]
but not understand that (r) represents the radius.
Solution: Learn what every variable represents.
14.2 Using the Wrong Formula
A question may provide similar-looking quantities but require a different mathematical relationship.
Solution: Identify what the question is asking before choosing a formula.
14.3 Mixing Units
For example, using a length in metres and another in centimetres without conversion can produce an incorrect result.
Solution: Convert measurements into compatible units first.
14.4 Calculator in the Wrong Mode
This is particularly common in trigonometry.
A calculator set to radians when the question expects degrees can produce an incorrect answer.
Solution: Check the calculator mode before solving trigonometric problems.
14.5 Rounding Too Early
Suppose your calculator produces:
[3.14159265]
and you immediately replace it with 3.14 during every stage of a long calculation.
The final answer may accumulate unnecessary rounding error.
Solution: Keep several decimal places during intermediate calculations and round at the end.
14.6 Confusing Radius and Diameter
Remember:
[
d=2r
]
Therefore:
[
r=\frac d2
]
Solution: Draw the circle and identify which measurement is given.
14.7 Forgetting Negative Signs
This is particularly common in algebra and the quadratic formula.
For example:
[
-b
]
means the negative of (b).
Solution: Write the values of (a), (b), and (c) separately before substitution.
15. How to Choose the Correct Mathematics Formula
One of the most important mathematical skills is recognizing which formula applies to a particular problem.
Use this five-step method.
Step 1: Read the question carefully
Do not start calculating immediately.
Step 2: Identify what is given
Write down the known values.
Step 3: Identify what is required
Determine exactly what the question asks you to find.
Step 4: Choose the formula
Look for a formula connecting the known values to the unknown quantity.
Step 5: Substitute and check
Substitute carefully, calculate, and check whether your answer makes sense.
Example
A rectangular classroom is 12 m long and 8 m wide. What is its area?
Given:
[
l=12m,\quad w=8m
]
Required: Area.
Formula:
[
A=lw
]
Substitution:
[
A=12\times8
]
Answer:
[
A=96m^2
]
This approach is more reliable than trying to guess the formula immediately.
16. Mathematics Formula Revision Checklist
Use this checklist when preparing for an examination.
- Percentages
- Percentage increase and decrease
- Average
- Ratio and proportion
- Speed, distance and time
- Profit and loss
- Simple interest
- Algebraic identities
- Laws of indices
- Quadratic formula
- Discriminant
- Distance formula
- Midpoint formula
- Gradient
- Equation of a straight line
- Circle equation
- Perimeter
- Area
- Circumference
- Arc length
- Sector area
- Pythagorean theorem
- Polygon angles
- Surface area
- Volume
- Sine
- Cosine
- Tangent
- Sine rule
- Cosine rule
- Trigonometric identities
- Arithmetic sequences
- Geometric sequences
- Arithmetic series
- Geometric series
- Mean
- Variance
- Standard deviation
- Z-score
- Probability
- Conditional probability
- Permutations
- Combinations
- Compound interest
- Logarithms
- Exponential growth and decay
- Differentiation
- Integration
17. Frequently Asked Questions About Mathematics Formulas
What are the most important mathematics formulas for students?
The most important formulas depend on your educational level and syllabus. However, students commonly need formulas covering percentages, algebra, geometry, trigonometry, statistics, probability, sequences, financial mathematics, and calculus.
How can I memorize mathematics formulas quickly?
Understand the meaning of each formula, organize formulas by topic, use active recall, create your own formula sheet, and practice solving problems. Memorization becomes easier when you repeatedly use a formula in different situations.
What is the most important mathematics formula?
There is no single formula that is most important for every student. Formulas such as the quadratic formula, Pythagorean theorem, area formulas, trigonometric ratios, probability rules, and statistical formulas are particularly useful depending on the subject and level.
What is the formula for speed?
The formula for speed is:
[
\text{Speed}=\frac{\text{Distance}}{\text{Time}}
]
If distance and time are known, divide distance by time to calculate speed.
What is the formula for distance?
[
\text{Distance}=\text{Speed}\times\text{Time}
]
For example, traveling at 60 km/h for 3 hours gives a distance of 180 km.
What is the formula for the area of a circle?
[
A=\pi r^2
]
where (r) is the radius of the circle.
What is the formula for the circumference of a circle?
[
C=2\pi r
]
where (r) is the radius.
What is the Pythagorean theorem?
For a right-angled triangle:
[
a^2+b^2=c^2
]
where (c) is the hypotenuse.
What is the quadratic formula?
For:
[
ax^2+bx+c=0
]
the solutions are:
[
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
]
What are the basic trigonometry formulas?
The three fundamental right-triangle ratios are:
[
\sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}
]
[
\cos\theta=\frac{\text{Adjacent}}{\text{Hypotenuse}}
]
[
\tan\theta=\frac{\text{Opposite}}{\text{Adjacent}}
]
They are commonly remembered using SOH-CAH-TOA.
What is the formula for simple interest?
[
I=PRT
]
where (P) is the principal, (R) is the interest rate expressed as a decimal, and (T) is the time.
What is the formula for compound interest?
[
A=P\left(1+\frac rn\right)^{nt}
]
The exact form used can depend on how frequently the interest is compounded.
What is the formula for standard deviation?
For a population:
[
\sigma=\sqrt{\frac{\sum(x-\mu)^2}{N}}
]
For a sample:
[
s=\sqrt{\frac{\sum(x-\bar{x})^2}{n-1}}
]
The distinction between population and sample calculations is important.
What is the difference between permutation and combination?
A permutation is used when order matters, while a combination is used when order does not matter.
Permutation:
[
{}_nP_r=\frac{n!}{(n-r)!}
]
Combination:
[
{}_nC_r=\frac{n!}{r!(n-r)!}
]
How do I know which mathematics formula to use?
First identify what the question gives you and what it asks you to find. Then choose a formula connecting those known and unknown quantities. Finally, substitute carefully and check the units and reasonableness of your answer.
18. Final Thoughts
Mathematics formulas are powerful tools, but successful mathematics students do more than memorize them.
They understand:
what the formula means,
when it should be used,
how to substitute values,
how to interpret the result,
and how to recognize and correct mistakes.
That is why this guide is designed as a reference rather than simply a formula list.
Use it while studying, but always combine formula revision with problem-solving practice. When you encounter a difficult formula, break it into smaller parts. Identify each variable, understand the relationship, work through a simple example, and then attempt a more challenging problem.
The goal is not to remember mathematics formulas for one examination and forget them afterward. The goal is to develop mathematical understanding that you can apply to new problems.
Remember:
Understand the formula. Practice the formula. Apply the formula. Check the answer.
With consistent practice, even a large collection of mathematics formulas becomes much easier to understand and use.
Suggested Further Reading for Students
To turn this page into a strong mathematics learning hub, consider creating detailed companion guides covering:
- Algebraic identities
- Quadratic equations
- Geometry formulas
- Circle theorems and formulas
- Trigonometry formulas
- Sine rule and cosine rule
- Statistics and standard deviation
- Probability
- Arithmetic and geometric sequences
- Financial mathematics
- Calculus for beginners
- Mathematics formulas for examination preparation
- Common mathematics mistakes students make
- How to study mathematics effectively
- Mathematics practice questions with solutions
These supporting resources can be internally linked to this comprehensive formula guide, creating a useful mathematics content hub for students.
Educational Note
Mathematical notation and the formulas required can vary according to a student’s level, curriculum, textbook, and examination syllabus. Students should use this guide alongside their official syllabus and approved learning materials. Where an examination provides an official formula sheet, that document should take precedence over any general online reference.
