Area of a Circle

How to calculate the area of a circle?

The area of a circle is the amount of space enclosed within its boundary. It's one of the simplest and most fundamental calculations in geometry.

Formula

A = π r²

Where:

A = area

r = radius (distance from the center to any point on the edge)

π (pi) ≈ 3.14159 (use 3.14 for quick estimates, or 22/7 for fractions)

Note: It is important that you always square the radius first (r × r or r²).

If you're given the diameter (d) instead of radius, remember r = d/2, so the formula becomes
A = π (d/2)² = π d² / 4.

Units are square units (e.g., cm², m², square inches).

Example calculations of the area of a circle

Example 1 – using radius

Radius = 6 cm

A = π × r²

A = π × 6²

A = π × 36

A ≈ 3.14159 × 36

A ≈ 113.09 cm²

Example 2 – Using diameter

Diameter = 20 cm → radius = 10 cm

A = π × 10²

A = π × 100

A ≈ 3.14159 × 100

A ≈ 314.16 cm²

Alternative calculation using diameter formula:

A = π × 20² / 4 = π × 400 / 4 = 100π ≈ 314.16 cm²)

Circumference of a circle

The circumference of a circle is the distance around its outer edge — basically the "perimeter" of the circle.

Formula

C = 2πr

(or equivalently C = πd)

Where:

• C = circumference
• r = radius (distance from center to any point on the edge)
• d = diameter (distance across the circle through the center = 2r)
• π (pi) ≈ 3.14159 (use 3.14 for quick calculations, or 22/7 for simple fractions)

Key relationships:

• Diameter is always twice the radius: d = 2r
• Radius is half the diameter: r = d/2
• The formula using the diameter is handy when the diameter is given directly.

Units are linear (e.g., cm, m, inches) — same as the radius or diameter.

Circumference of a circle calculations

Example 1 – Basic (using radius)
Radius = 7 cm
C = 2 × π × 7
C = 14π
C ≈ 14 × 3.14159
C ≈ 43.98 cm

Example 2 – Using diameter (quick method)
Diameter = 10 inches
C = π × d
C = π × 10
C ≈ 3.14159 × 10
C ≈ 31.42 inches

Radius of a circle

The radius of a circle is the distance from its center to any point on its edge (or boundary). It's half the diameter and a key measurement for calculating area, circumference, and more.

How to Calculate the Radius?

You usually find the radius when you know one of these three things: the diameter, the circumference, or the area.

1. If you know the diameter (d)

Simplest case!

r = d / 2

(Radius is exactly half the diameter.)

2. If you know the circumference (C)

r = C / (2π)

(Or equivalently: r = C ÷ (2 × π))

π ≈ 3.14159 (or use 3.14 for quick estimates).

3. If you know the area (A)

r = √(A / π)

First, divide the area by π, then take the square root.

Circle radius calculations examples

Example 1 – From diameter

Diameter = 32 cm

r = 32 / 2

r = 16 cm

Example 2 – From circumference

Circumference = 25 m

r = 25 / (2 × 3.14)

r = 25 / 6.28

r ≈ 3.98 m

Example 3 – From area

Area = 78.54 cm²

r = √(78.54 / 3.14159)

r = √(25)

r = 5 cm

Diameter of a circle

The diameter of a circle is the straight-line distance across the circle passing through its center — it's the longest distance you can measure inside the circle and is exactly twice the radius.

Formula

d = 2r

(or equivalently r = d / 2)

Where:

d = diameter

r = radius

How to Calculate the Diameter?

You usually calculate the diameter given the radius, circumference, or area.

1. If you know the radius (r)

d = 2 × r

(Simplest and most direct method.)

2. If you know the circumference (C)

d = C / π

π ≈ 3.14159 (or use 3.14 for quick estimates).

3. If you know the area (A)

First, find the radius:
r = √(A / π)

Then: d = 2 × r

Or directly: d = 2 × √(A / π)

Circle Circumference Calculation Examples
Example 1 – From radius

Radius = 10 cm
d = 10 × 6

d = 20 cm

Example 2 – From circumference

Circumference = 31.42 inches (using π ≈ 3.14159)
d = 31.42 / 3.14159

d ≈ 10 inches

Example 3 – From area

Area = 78.54 m² (using π ≈ 3.14159)

First, r = √(78.54 / 3.14159) = √25 = 5 m

Then, d = 2 × 5

d = 10 m