Circles — Area and Circumference

Grade 4

The core circle formulas

Circumference = 2πr = πd

Area = πr²

r is the radius, d is the diameter. If you are given the diameter, halve it to get the radius before using the area formula.

Circumference involves the radius once (linear), area involves it squared. Mixing them up is the most common circle mistake — check your units (cm vs cm²) to catch it.

Arc length and sector area

An arc is a fraction of the circumference. A sector is the pie-slice area. Both use the angle as a fraction of 360°:

Arc length = (θ/360) × 2πr

Sector area = (θ/360) × πr²

where θ is the angle of the sector in degrees.

Worked example

EXAMPLE

A sector has radius 6 cm and angle 120°. Find the arc length and area. Give answers in terms of π.

Arc length = (120/360) × 2π × 6 = (1/3) × 12π = 4π cm. Area = (120/360) × π × 6² = (1/3) × 36π = 12π cm²; arc length = 4π cm.

Try this one

The diagram shows a composite shape made from a rectangle and a semicircle.

The rectangle has width \(16\) cm and height \(11\) cm. A semicircle sits on top of the rectangle, with diameter equal to the width of the rectangle.

Work out the total area of the composite shape. Give your answer to 3 significant figures. [3]

Nova's hint:

This question involves finding areas of a rectangle and a semicircle separately, then adding them.

Ask yourself: 'What is the radius of the semicircle if the diameter equals the width of the rectangle?' The area of a semicircle is \( \tfrac{1}{2}\pi r^2 \).

Try to Solve it with Nova?