Perimeter Problems Including Arcs

Grade 5

Perimeter of a sector

The perimeter of a sector is not just the arc — it includes the two radii as well.

Sector perimeter = arc length + 2r

It is easy to forget the straight edges. Slow down, sketch the shape, and trace the full boundary.

Compound shapes with arcs

When a shape mixes straight sides with semicircles or quarter-circles, work out each piece separately: straight edges by measurement, curved edges using the arc length formula, then add everything.

A semicircle's perimeter contribution is πr (half the circumference), not the diameter. Add the diameter separately if it forms part of the straight boundary.

Worked example

EXAMPLE

A running track has two straight sections each 80 m long and two semicircular ends each with diameter 60 m. Find the total perimeter of the track.

Each semicircle has radius 30 m. Circumference of one full circle = 2π × 30 = 60π m. Two semicircles = one full circle = 60π m. Two straight sections = 2 × 80 = 160 m. Total perimeter = 160 + 60π ≈ 160 + 188.5 = 348.5 m (to 1 d.p.).

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?