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Circle: Circumference, Pi and Area

The distance around a circle is its circumference: C = π × d = 2πr. The space inside is its area: A = πr². Pi (π) is the number you get when you divide the circumference by the diameter. It is about 3.14 for every circle, big or small.

🎬 Step-by-step story

  1. This is a round disc. The green line is the radius r, from centre to edge. The yellow line is the diameter d. d is two radii.
  2. Roll the disc one full turn on a ruler. The red mark draws a red line. Its length is the distance around the disc: the circumference.
  3. Lay diameter sticks along the red line. Three whole sticks fit, and a small bit is left. That bit is 0.14 of a stick. So the circumference is 3.14 diameters.
  4. Now the inside. Cut the circle into 16 equal slices, like a pizza. We will not throw anything away.
  5. Stack the slices up and down. They make almost a rectangle. Its length is half the rim (π × r) and its height is r. So area = π × r × r.
  6. Your turn. Change the radius. Press Cut and Join. Watch the circumference and the area numbers change.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

What is the difference between radius and diameter?

The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii.

Is circumference a length or an area?

It is a length: the distance around. The red line in the 3D is straight and has only a length. Area is the space inside.

Why is pi 3.14 and not exactly 3?

Three sticks do not quite reach the end of the red line. A small bit is left. That extra bit is about 0.14 of a stick.

Is pi exactly 22/7?

22/7 is 3.1428... and pi is 3.14159... They are close but not equal. Pi never ends, so we use a close value.

Why does the area have r times r?

Area is a flat space, so it needs two lengths. The rectangle from the slices has length π × r and height r.

Do the slices really make a rectangle?

Almost. With 16 slices the edges are bumpy. With more and more slices, it gets closer to a perfect rectangle.

Does pi change for a bigger circle?

No. Change the radius slider: C and A change, but circumference ÷ diameter stays 3.14.

The parts of a circle

A circle is a round shape. Every point on its edge is the same distance from the middle point, the centre.

Length is measured in cm or m. Area is measured in square units: cm² or m².

What is pi (π)?

Take any circle. Measure the distance around it. Measure the width across it. Now divide: circumference ÷ diameter. You always get about 3.14. This special number is called pi, written with the Greek letter π.

It does not matter if the circle is a coin or a big wheel. The rim is always about 3.14 times the width. That is why 3 sticks and a bit fit along the rim.

Pi is a number whose digits never end and never repeat: 3.14159265... In school we use 3.14 or the fraction 22/7. Both are close, but not exact. Use 22/7 when the radius is a multiple of 7. Use 3.14 when the question says so.

Circumference of a circle

Since C ÷ d = π, we get:

C = π × d

The diameter is two radii, so we can also write:

C = 2 × π × r

Example: r = 7 cm. C = 2 × 22/7 × 7 = 44 cm.

If you know C and want the radius, work backwards: r = C ÷ (2π).

Area of a circle

To find the area, cut the circle into many equal thin slices, like a pizza. Put them side by side, one pointing up and the next pointing down. They fit together into a shape that is almost a rectangle.

Area of rectangle = length × height = π × r × r. So:

A = π × r²

The more slices you cut, the straighter the shape becomes. Our 3D shows 16 slices.

Using the formulas well

  1. Find out what is given: radius or diameter? If it is the diameter, halve it first for the area.
  2. Choose the formula: distance around gives C, space inside gives A.
  3. Put in π as 22/7 or 3.14, as the question says.
  4. Write the unit: cm for C, cm² for A.

If the radius doubles, C doubles but A becomes 4 times bigger, because r is squared.

Try it

At home: Take a round plate or a bangle. Wrap a string once around it and mark the length. Lay the string straight and measure it with a ruler. Now measure the width across the plate. Divide the string length by the width. Do you get a number near 3.14? Try a bottle cap and a bucket too.

In the 3D: Before you press Roll, guess how many diameter sticks will fit. Then check.

Key formulas and definitions

Worked examples

1. A circle has radius 7 cm. Find its circumference. (π = 22/7)

C = 2πr = 2 × 22/7 × 7 = 44 cm.

2. A round table top has diameter 10 cm. Find its circumference. (π = 3.14)

C = πd = 3.14 × 10 = 31.4 cm.

3. A circle has radius 7 cm. Find its area. (π = 22/7)

A = πr² = 22/7 × 7 × 7 = 22 × 7 = 154 cm².

4. A wheel has diameter 70 cm. How far does it go in 10 turns? (π = 22/7)

One turn = circumference = 22/7 × 70 = 220 cm. Ten turns = 220 × 10 = 2200 cm = 22 m.

5. The circumference of a circle is 66 cm. Find its radius and its area. (π = 22/7)

r = C ÷ 2π = 66 ÷ (2 × 22/7) = 66 × 7 ÷ 44 = 10.5 cm. A = 22/7 × 10.5 × 10.5 = 346.5 cm².

6. A round park has radius 14 m. Fencing costs ₹10 per metre. Find the cost of the fence and the area of the park. (π = 22/7)

C = 2 × 22/7 × 14 = 88 m. Cost = 88 × 10 = ₹880. A = 22/7 × 14 × 14 = 616 m².

7. A flat ring has outer radius 10 cm and inner radius 6 cm. Find its area. (π = 3.14)

Ring = big circle minus small circle = π × 10² − π × 6² = 3.14 × (100 − 36) = 3.14 × 64 = 200.96 cm².

Common mistakes

Practice quiz

1. The distance around a circle is called its:
2. Circumference ÷ diameter is always about:
3. The area of a circle of radius 3 cm (π = 3.14) is about:
4. If the diameter is 12 cm, the radius is:
5. If the radius is doubled, the area becomes:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is pi in simple words?

Pi is how many times the width of a circle fits around its edge. It is about 3.14 for every circle.

Should I use 22/7 or 3.14 for pi?

Do what the question says. If it does not say, use 22/7 when the radius is a multiple of 7, otherwise 3.14. Both are only close values.

Why is the area of a circle πr² and not 2πr?

Area is a flat space, so it needs two lengths multiplied together: π × r (half the rim) and r. 2πr is the length around, not the space inside.

Where this is taught

ItalyScuola secondaria di primo grado – classe 3ªSpace and figures

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