Pi is the ratio of any circle's circumference to its diameter, about 3.14159. It is the same for every circle.

How do you find the area of a circle from the diameter?

Halve the diameter to get the radius, then use A = π r^2.

Circles and Pi: Circumference and Area of a Circle

The parts of a circle, what pi really is, where the circumference and area formulas come from (including a slice-and-rearrange picture of $\pi r^2$ ), and arcs and sectors.

3 min read Supports 4 lessons

A circle is every point at the same distance from a centre. That distance is the radius, $r$ . Twice the radius, straight across through the centre, is the diameter, $d$ . The distance all the way round is the circumference, $C$ .

What pi is

Measure any circle, a coin, a plate, a running track, and divide its circumference by its diameter. You always get the same number, a little over $3$ . That number is $\pi$ .

\pi = \frac{C}{d} \approx 3.14159\ldots

$\pi$ is irrational: its decimals never end or repeat.

Drag the radius and watch both measurements change together.

Interactive

C = 2\pi r = 4\pi \approx 12.57

A = \pi r^2 = 4\pi \approx 12.57

Radius r2

Circle formulas

C = 2\pi r = \pi d \qquad A = \pi r^2

Why the area is πr²

Cut a circle into many thin slices like a pizza, and lay them out alternately point-up and point-down. They form a shape that is almost a rectangle. The more slices, the closer it gets.

$\text{Cut the circle into thin sectors.}$

1/4

Watch out

Area uses the radius. If you are given a diameter, halve it first. A circle with diameter $10$ has area $25\pi$ , not $100\pi$ .

Worked example

A circular table has diameter $1.2$ m. Find its area and circumference.

1
The radius is half the diameter.
$r = 0.6$

Tip

Leave answers in terms of $\pi$ , like $0.36\pi$ , when an exact value is wanted.

Arcs and sectors

An arc is part of the circumference, and a sector is a slice of the area. A $90°$ sector is a quarter of the circle, so it has a quarter of the area. In general, for an angle $\theta$ :

\text{arc length} = \frac{\theta}{360^\circ} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360^\circ} \times \pi r^2

Key idea

Measuring angles in radians makes these much simpler: arc length becomes $r\theta$ . That is one reason calculus uses radians.

Common questions

What exactly is pi?: Pi is the ratio of any circle's circumference to its diameter, about $3.14159$ . It is the same for every circle.
How do you find the area of a circle from the diameter?: Halve the diameter to get the radius, then use $A = \pi r^2$ .

Practise circles and pi

The Circles lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Geometry

Circles and Pi: Circumference and Area of a Circle

The parts of a circle, what pi really is, where the circumference and area formulas come from (including a slice-and-rearrange picture of $\pi r^2$ ), and arcs and sectors.

3 min read Supports 4 lessons

What pi is

Measure any circle, a coin, a plate, a running track, and divide its circumference by its diameter. You always get the same number, a little over $3$ . That number is $\pi$ .

\pi = \frac{C}{d} \approx 3.14159\ldots

$\pi$ is irrational: its decimals never end or repeat.

Drag the radius and watch both measurements change together.

Interactive

C = 2\pi r = 4\pi \approx 12.57

A = \pi r^2 = 4\pi \approx 12.57

Radius r2

Circle formulas

C = 2\pi r = \pi d \qquad A = \pi r^2

Why the area is πr²

Cut a circle into many thin slices like a pizza, and lay them out alternately point-up and point-down. They form a shape that is almost a rectangle. The more slices, the closer it gets.

$\text{Cut the circle into thin sectors.}$

1/4

Watch out

Area uses the radius. If you are given a diameter, halve it first. A circle with diameter $10$ has area $25\pi$ , not $100\pi$ .

Worked example

A circular table has diameter $1.2$ m. Find its area and circumference.

1
The radius is half the diameter.
$r = 0.6$

Tip

Leave answers in terms of $\pi$ , like $0.36\pi$ , when an exact value is wanted.

Arcs and sectors

An arc is part of the circumference, and a sector is a slice of the area. A $90°$ sector is a quarter of the circle, so it has a quarter of the area. In general, for an angle $\theta$ :

\text{arc length} = \frac{\theta}{360^\circ} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360^\circ} \times \pi r^2

Key idea

Measuring angles in radians makes these much simpler: arc length becomes $r\theta$ . That is one reason calculus uses radians.

Common questions

What exactly is pi?: Pi is the ratio of any circle's circumference to its diameter, about $3.14159$ . It is the same for every circle.
How do you find the area of a circle from the diameter?: Halve the diameter to get the radius, then use $A = \pi r^2$ .

Practise circles and pi

The Circles lesson has worked examples and unlimited practice that adapts to you.

Open the lesson