Circle Calculations | Cambridge IGCSE Mathematics

Circle Calculations Guide

Cambridge IGCSE Mathematics - Calculations Involving Circles

Key Circle Formulas

Circumference

C = 2πr

C = πd

Where r is radius, d is diameter

Area of Circle

A = πr²

Where r is radius

Arc Length

L = (θ/360) × 2πr

Where θ is the angle in degrees

Sector Area

A = (θ/360) × πr²

Where θ is the angle in degrees

Worked Examples

Example 1: Sector Perimeter

Problem: The perimeter of a sector with radius 8 cm is 26 cm. Find the angle.

Solution:

Perimeter = 2r + arc length

26 = 2×8 + (θ/360)×2π×8

26 = 16 + (θ/360)×16π

10 = (θ/360)×16π

θ = (10 × 360)/(16π) ≈ 71.6°

Example 2: Semicircular Container

Problem: Semicircular container with radius 0.45 m and length 4 m. Find volume.

Solution:

Cross-sectional area = ½ × π × (0.45)² ≈ 0.318 m²

Volume = area × length = 0.318 × 4 = 1.272 m³

Example 3: Paper Cylinder

Problem: Rectangular sheet 28 cm × 20 cm formed into cylinder. Find radius.

Solution:

The length becomes the circumference: 28 = 2πr

r = 28/(2π) ≈ 4.46 cm

Problem-Solving Strategies

1. Identify What's Given

Always start by identifying what information is provided: radius, diameter, circumference, area, angle, etc.

2. Choose the Right Formula

Based on what's given and what needs to be found, select the appropriate formula.

3. Unit Consistency

Ensure all measurements are in the same units before calculating. Convert if necessary.

4. Work Step by Step

Break down complex problems into smaller steps. Show your working clearly.

Practice Quiz

Question 1

Calculate the circumference of a circle with radius 4.7 cm.

Question 2

A circle has a radius of 18 cm. Calculate its circumference.

Question 3

A circular disc has circumference 250 cm. Calculate its radius.

Question 4

Calculate the area of a sector with radius 4.8 cm and sector angle 45°.

Question 5

The radius of a circle is 42 cm. Work out the circumference in terms of π.

Quiz Answers

C = 2πr = 2 × π × 4.7 ≈ 29.53 cm
C = 2πr = 2 × π × 18 ≈ 113.10 cm
r = C/(2π) = 250/(2π) ≈ 39.79 cm
A = (θ/360) × πr² = (45/360) × π × (4.8)² ≈ 9.05 cm²
C = 2πr = 2 × π × 42 = 84π cm

Common Problem Types

Problem Type	Key Approach	Example
Finding circumference	Use C = 2πr or C = πd	Given r = 5 cm, C = 10π cm
Finding area	Use A = πr²	Given r = 7 cm, A = 49π cm²
Sector problems	Use fraction of circle (θ/360)	60° sector: area = (60/360) × πr²
Arc length	L = (θ/360) × 2πr	90° arc: L = (90/360) × 2πr = ½πr
Reverse problems	Rearrange formulas	Given C, find r: r = C/(2π)

Exam Tips

Always include units in your final answer
When asked to give an answer in terms of π, don't multiply by π
For sector problems, remember to add the two radii to the arc length for perimeter
Check if the problem involves a full circle, semicircle, or sector
For compound shapes, break them down into simpler parts