Circles and Circumference

Identify parts of a circle, understand π and calculate the distance around circular objects.

The distance around a circle

Coins, wheels, plates and round water tanks all have circular boundaries. The perimeter of a circle has a special name: its circumference.

What You Will Learn

• Identify the radius, diameter, circumference, arc and sector.
• Investigate the relationship between circumference and diameter.
• Explain the meaning and approximate value of π.
• Use C = πd and C = 2πr.

1. Parts of a Circle

• Centre: the fixed point in the middle.
• Radius (r): a segment from the centre to the circumference.
• Diameter (d): a segment through the centre joining two points on the circumference.
• Circumference (C): the complete distance around the circle.
• Arc: part of the circumference.
• Sector: the region enclosed by two radii and an arc.

d = 2r   and   r = d ÷ 2

2. Investigating Circumference and Diameter

Wrap string once around a circular object and measure the string to find C. Measure straight across the centre to find d. Then calculate C ÷ d.

The quotient is nearly constant for every circle:

C ÷ d = π

3. Meaning of π

The symbol π, read “pi”, represents the constant ratio of a circle’s circumference to its diameter.

π ≈ 3.142   or   π ≈ 22/7

π is not exactly equal to either approximation. Use the value requested in the question. The fraction 22/7 is convenient when a measurement is a multiple of 7.

4. Deriving the Circumference Formulae

Since C ÷ d = π, multiplying both sides by d gives:

C = πd

Since d = 2r, replace d with 2r:

C = π(2r) = 2πr

Use C = πd when the diameter is known. Use C = 2πr when the radius is known.

5. Worked Examples

Example 1: Diameter given

A circle has diameter 14 cm. Using π = 22/7:
C = πd = 22/7 × 14 = 44 cm.

Example 2: Radius given

A wheel has radius 10 cm. Using π = 3.142:
C = 2πr = 2 × 3.142 × 10 = 62.84 cm.

Example 3: Finding diameter

A circular pond has circumference 31.42 m. Using π = 3.142:
d = C ÷ π = 31.42 ÷ 3.142 = 10 m.

Example 4: Wheel revolutions

A wheel’s circumference is 2.2 m. In 50 complete revolutions, distance = 50 × 2.2 = 110 m.

Common Errors

• Confusing radius with diameter.
• Using 2πd instead of πd.
• Forgetting units in the final answer.
• Using square units for circumference.
• Rounding too early during a calculation.

Let’s Practise

1. Find the diameter of a circle with radius 6.5 cm.
2. Find the radius of a circle with diameter 24 m.
3. Using π = 22/7, find the circumference when d = 21 cm.
4. Using π = 3.142, find the circumference when r = 8 cm.
5. A circle has circumference 62.84 cm. Find its diameter using π = 3.142.
6. Find the radius of a circle with circumference 44 m using π = 22/7.
7. A bicycle wheel has diameter 70 cm. How far does it travel in one revolution? Use π = 22/7.
8. The same wheel makes 100 revolutions. Find the distance in metres.
9. Name the region enclosed by two radii and an arc.
10. Explain why C ÷ d is nearly the same for every circle.

Lesson Summary

The circumference is the distance around a circle. The constant ratio C ÷ d is π. Therefore, C = πd and, because d = 2r, C = 2πr.

Practice Answers: 1. 13 cm | 2. 12 m | 3. 66 cm | 4. 50.272 cm | 5. 20 cm | 6. 7 m | 7. 220 cm | 8. 220 m | 9. Sector | 10. All circles are similar, so circumference and diameter increase in the same ratio, π