⭕Circles

Wheels are circles, plates are circles, the sun looks like a circle. It is the most common shape in nature and engineering, and it has a remarkable property: one single number — the radius — determines everything about it.

Radius, diameter, center

• Center: the middle point of the circle, usually called $O$
• Radius $r$: the distance from the center to any point on the circle
• Diameter $d$: a chord through the center with both ends on the circle, $d = 2r$

Every point on the circle is the same distance from the center — that is literally the definition of a circle, and the reason wheels roll smoothly.

Radius r3

Diameter d

d = 6

Circumference C = 2πr

C = 18.85

Area A = πr²

A = 28.27

C = 2πr = 2π×3 ≈ 18.85 · A = πr² = π×3² ≈ 28.27

Circumference and π

The distance around the circle is the circumference $C$. Divide any circle's circumference by its diameter and you always get the same number:

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

That is pi — a decimal that never repeats. So:

$C = 2\pi r = \pi d$

Where the area formula comes from

The area of a circle is:

$A = \pi r^2$

Why? Slice the circle into many thin wedges and rearrange them. They fit into an approximate rectangle: about $\pi r$ wide (half the circumference) and $r$ tall (the radius). So the area is $\pi r \times r = \pi r^2$. The thinner the slices, the more rectangular the result.

Worked example

A circle of radius 3 cm:

• diameter $d = 6$ cm
• circumference $C = 2\pi \times 3 = 6\pi \approx 18.85$ cm
• area $A = \pi \times 3^2 = 9\pi \approx 28.27$ cm²

Check yourself

✅

Quick quiz

↺ Try again

1. 1. A circle has diameter 10. What is its radius?

   51020

2. 2. Which formula gives the circumference?

   πr2πrπr²

3. 3. If the radius doubles, the area becomes…

   2×4×8×