Circles: Circumference and Area

The shape with no corners

A circle is the set of all points the same distance from a single centre point. That simple idea gives the circle its perfect roundness — wheels, clock faces, coins and pizzas are all circles. Because a circle has no straight sides, we cannot measure it the way we measure rectangles. We need special formulas, and a very famous number called pi.

In this lesson you will learn the parts of a circle, how to find the distance around it (the circumference) and the space inside it (the area). If you have measured straight-sided shapes before, area and perimeter is good background.

The parts of a circle

Part	What it is
Centre	The middle point, the same distance from every edge point
Radius (r)	The distance from the centre to the edge
Diameter (d)	The distance straight across, through the centre
Circumference (C)	The total distance around the outside

The most important relationship to remember is:

Diameter = 2 × radius, and so radius = diameter ÷ 2.

The diameter is just two radii placed end to end across the middle. Get this link wrong and every other answer will be wrong, so always check which one a question gives you.

Pi: the magic number of circles

Here is something remarkable. Take any circle — tiny or huge — and divide its circumference by its diameter. You always get the same number: about 3.14159.... This never-ending, never-repeating number is called pi, written with the Greek letter π.

π = circumference ÷ diameter ≈ 3.14

In plain words, the diameter fits around the circle a little more than 3 times — about 3.14 times, to be exact. That is true for the rim of a cup and for the orbit of a planet. For most schoolwork we use π ≈ 3.14, though calculators have a more accurate π button.

Circumference: the distance around

Rearranging π = C ÷ d gives the circumference formula:

Circumference = π × diameter, or equivalently C = 2 × π × radius

Both versions are the same formula, because the diameter equals 2 × radius. Use whichever matches the number you are given.

Worked example 1 — given the diameter

A bicycle wheel has a diameter of 60 cm. Find its circumference (π ≈ 3.14).

Step 1 — Choose the formula. We have the diameter, so use C = π × d. Step 2 — Substitute: C = 3.14 × 60 Step 3 — Multiply: C = 188.4 cm

So the wheel travels 188.4 cm in one full turn — a neat fact for working out distances cycled.

Worked example 2 — given the radius

A circular pond has a radius of 7 m. Find its circumference.

Step 1 — Choose the formula. We have the radius, so use C = 2 × π × r. Step 2 — Substitute: C = 2 × 3.14 × 7 Step 3 — Multiply step by step: 2 × 3.14 = 6.28, then 6.28 × 7 = 43.96 m.

(If you prefer, first double the radius to get the diameter, 14 m, then do 3.14 × 14 = 43.96 m. Same answer.)

Area: the space inside

The space inside the circle uses a different formula:

Area = π × radius² (often written A = πr²)

Notice it uses the radius, not the diameter, and the radius is squared (multiplied by itself). Because area is two-dimensional, the answer is always in square units like cm².

Worked example 3 — straightforward area

Find the area of a circle with radius 4 cm (π ≈ 3.14).

Step 1 — Square the radius first: 4² = 4 × 4 = 16 Step 2 — Multiply by π: 3.14 × 16 = 50.24 cm²

A very common mistake is to do (3.14 × 4)² or to multiply π by the radius before squaring. Always square the radius first, then multiply by π.

Worked example 4 — given the diameter (a trap!)

A circular table has a diameter of 10 cm. Find its area.

Step 1 — Find the radius first. The formula needs the radius: r = diameter ÷ 2 = 10 ÷ 2 = 5 cm. Step 2 — Square the radius: 5² = 25 Step 3 — Multiply by π: 3.14 × 25 = 78.5 cm²

If you had used 10 by mistake you would get a wildly wrong answer, so always convert the diameter to the radius before using the area formula.

Comparing the two formulas

It is easy to mix these up, so keep them side by side:

Want	Formula	Uses	Units
Circumference (around)	C = πd or 2πr	diameter or radius	cm (length)
Area (inside)	A = πr²	radius, squared	cm² (square)

A memory hook many students use: "Cherry pie's delicious" for C = πd (Circumference = pi × diameter), and "Apple pies are too" for A = πr² (Area = pi × r²).

Why does πr² give the area?

Here is the idea, without heavy proof. Imagine slicing a circle into many thin wedges, like a pizza, then unrolling and rearranging them so the wedges alternate point-up and point-down. They form a shape very close to a rectangle. That rectangle's height is the radius r, and its width is half the circumference, πr. So its area is height × width = r × πr = πr². The more slices you use, the closer it gets to a perfect rectangle — which is why the circle's area really is πr². This connects circles back to the rectangles you already know. To carry these skills into 3D, see surface area and volume.

Practice activity

Hunt for circles around your home and measure them.

1. Find three circular objects — a plate, a tin lid, a coin.
2. Wrap a piece of string around each one, then measure the string with a ruler: that is the circumference. Measure straight across the middle for the diameter.
3. Divide your circumference by your diameter for each object. You should get a number close to 3.14 every time — you have discovered pi yourself!
4. Using the radius (half the diameter), calculate each object's area with A = πr², and write a sentence on whether circumference or area would tell you how much wrapping ribbon you need.

Quick recap

A circle is defined by its centre and radius, with the diameter being twice the radius. Use C = πd (or 2πr) for the distance around, and A = πr² for the space inside, always squaring the radius first. The constant pi, about 3.14, is the unchanging link between a circle's diameter and its circumference.