How to find the area of a circle, triangle, and rectangle
Area is one of the most practical concepts in geometry. Whether you are figuring out how much tile you need for a floor, how large a garden plot is, or solving a problem on a math test, knowing how to find the area of basic shapes saves you every time.
This guide covers the three shapes that come up most often: circles, triangles, and rectangles. For each one, you will get the formula, a clear explanation of what each part means, worked examples, and answers to the variations that trip people up, like half circles, right triangles, and rectangles with fractions.
What is area?
Area is the amount of flat space inside a two-dimensional shape. It is always expressed in square units, such as square centimetres (cm²), square metres (m²), or square inches (in²). When you calculate the area of a shape, you are measuring how much surface it covers.
How to find the area of a circle
The formula:
Area of circle = π × r²
Where:
r is the radius (the distance from the centre of the circle to its edge)
π (pi) is approximately 3.14159
If you are given the diameter instead of the radius, divide it by 2 first. The radius is always half the diameter.
Step-by-step example
A circular garden has a radius of 5 metres. What is its area?
Write the formula: A = π × r²
Substitute the radius: A = π × 5²
Square the radius: A = π × 25
Multiply: A = 3.14159 × 25
Answer: A ≈ 78.54 m²
How to find the area of a half circle
A semicircle is exactly half a full circle, so you use the same formula and divide by 2.
Area of a half circle = (π × r²) ÷ 2
Example: A semicircle with a radius of 6 cm.
A = (π × 6²) ÷ 2 = (π × 36) ÷ 2 = 113.1 ÷ 2 = 56.55 cm²
How to find the area of a quarter circle
A quarter circle is one-fourth of a full circle.
Area of a quarter circle = (π × r²) ÷ 4
Example: Quarter circle with radius 8 cm.
A = (π × 64) ÷ 4 = 201.06 ÷ 4 = 50.27 cm²
How to find the area of a sector
A sector is a "pie slice" of a circle defined by an angle.
Area of a sector = (θ ÷ 360) × π × r²
Where θ is the angle of the sector in degrees.
Example: A sector with radius 10 cm and an angle of 90°.
A = (90 ÷ 360) × π × 100 = 0.25 × 314.16 = 78.54 cm²
Note: A 90° sector is the same as a quarter circle, which confirms the formula works correctly.
Circumference vs. area: What's the difference?
People often confuse area and circumference. Area measures the space inside the circle. Circumference measures the distance around it.
Area = π × r²
Circumference = 2 × π × r
Both use the radius, but they measure completely different things.
How to find the area of a triangle
The formula:
Area = ½ × base × height
Where:
base is any one side of the triangle
height is the perpendicular distance from that base to the opposite vertex (the height must be at a right angle to the base)
Step-by-step example
A triangle has a base of 10 cm and a height of 6 cm.
Write the formula: A = ½ × b × h
Substitute: A = ½ × 10 × 6
Multiply: A = ½ × 60
Answer: A = 30 cm²
How to find the area of a right triangle
A right triangle has one 90° angle. The two sides that form the right angle are called the legs. You use the same formula, treating the two legs as the base and height.
Area = ½ × leg₁ × leg₂
Example: A right triangle with legs of 8 cm and 5 cm.
A = ½ × 8 × 5 = ½ × 40 = 20 cm²
How to find the area of a triangle without the height
If you know all three sides but not the height, use Heron's Formula.
Steps:
Find the semi-perimeter: s = (a + b + c) ÷ 2
Apply the formula: A = √(s × (s−a) × (s−b) × (s−c))
Example: A triangle with sides 5 cm, 6 cm, and 7 cm.
s = (5 + 6 + 7) ÷ 2 = 9
A = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 cm²
How to find the area of a triangle using coordinates
If you have three points on a coordinate grid (x₁, y₁), (x₂, y₂), (x₃, y₃), use the coordinate formula:
A = ½ × |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
The absolute value (| |) ensures you always get a positive result.
How to find the area of an obtuse triangle
An obtuse triangle has one angle greater than 90°. The base-times-height formula still works, but you need to make sure the height is measured correctly: it must drop perpendicularly from the opposite vertex to the line containing the base, which sometimes falls outside the triangle itself.
The formula does not change. Only how you identify the height changes.
How to find the area of a rectangle
The formula:
Area = length × width
Also written as A = l × w. Both dimensions must be in the same unit before multiplying.
Step-by-step example
A rectangle is 12 cm long and 4 cm wide.
Write the formula: A = l × w
Substitute: A = 12 × 4
Answer: A = 48 cm²
How to find the area and perimeter of a rectangle
These are two different measurements:
Area = length × width (space inside)
Perimeter = 2 × (length + width) (distance around the outside)
Example: A rectangle 9 m long and 3 m wide.
Area = 9 × 3 = 27 m²
Perimeter = 2 × (9 + 3) = 2 × 12 = 24 m
How to find the area of a rectangle with fractions
The process is identical. Multiply the fractional length by the fractional width.
Example: A rectangle that is 2½ cm × 1¾ cm.
Convert: 2½ = 5/2 and 1¾ = 7/4
A = 5/2 × 7/4 = 35/8 = 4.375 cm²
How to find the area of a rectangle with variables
If one or both dimensions include variables, multiply the expressions together.
Example: length = (x + 3) and width = (x − 1)
A = (x + 3)(x − 1) = x² − x + 3x − 3 = x² + 2x − 3
How to find the missing side of a rectangle when you know the area
If you know the area and one side, divide area by the known side.
Missing side = Area ÷ known side
Example: Area = 72 cm², width = 8 cm. Length = 72 ÷ 8 = 9 cm
Quick reference: All three formulas
Shape | Formula | What You Need |
|---|---|---|
Circle | A = π × r² | Radius |
Half Circle | A = (π × r²) ÷ 2 | Radius |
Quarter Circle | A = (π × r²) ÷ 4 | Radius |
Triangle | A = ½ × base × height | Base and height |
Triangle (no height) | A = √(s(s−a)(s−b)(s−c)) | All three sides |
Rectangle | A = length × width | Length and width |
Common mistakes to avoid
Using diameter instead of radius in the circle formula. Always halve the diameter first.
Forgetting to square the radius. The formula is π × r², not π × r.
Using the wrong height for a triangle. Height must be perpendicular to the base, not a slanted side.
Mixing units. If length is in centimetres and width is in metres, convert one before multiplying.
Forgetting the ½ in the triangle formula. A triangle is half a rectangle, so the formula is always half of base times height.
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For students building confidence across all areas of geometry, understanding how percentages, fractions, and shape calculations connect is part of the bigger picture. If you are also working on topics like how to calculate percentages quickly without mistakes or converting fractions to decimals, those skills feed directly into area problems that involve fractional measurements.