The length of an arc and the perimeter of a sector

Home learning focus

Learn how to find the length of an arc and the perimeter of a sector.

This lesson includes:

  • learning summary
  • two activity sheets

Learn

Students looking to achieve grade 4 in GCSE Maths must understand how to calculate the length of an arc and the perimeter of a sector.

Read page 5 of our 'Circles, sectors and arcs' Bitesize revision guide and page 3 from our 'Sector, segment and arc' Bitesize revision guide to understand:

Revise - length of an arc

Arc length

An arc is a portion of the circumference of a circle.

A chord separates the circumference of a circle into two sections - the major arc and the minor arc. It also separates the area into two segments - the major segment and the minor segment.

Worked example

Calculate the arc length to 2 decimal places (dp).

First calculate what fraction of a full turn the angle is.

The angle is 90°, so it is one quarter of the whole circle (360°). The arc length is ¼ of the full circumference.

The circumference of a circle is πd (or π × diameter) and the diameter is = 2 × radius.

Remember that π is = 3.14 (to 2 dp)

The formula to calculate the arc length is: Arc length = angle ÷ 360° × π × d

The arc length is: ¼ × π × 8 = 2π

is 2 × 3.14

The arc length is 6.28 cm (to 2 dp).

Perimeter of a sector

The perimeter is the distance all around the outside of a shape. We can find the perimeter of a sector using what we know about finding the length of an arc.

A sector is formed between two radii and an arc. To find the perimeter, we need to add these values together.

Perimeter = Arc length + 2r

Worked example

In this diagram, the arc length (27.5 cm) and the radius (45 cm) are shown.

From this the perimeter can be calculated:

Perimeter = 27.5 + (2 × 45)

27.5 + 90 = 117.5 cm

The perimeter is 117.5 cm.

To learn more about arcs look at the Circles, sectors and arcs Bitesize guide here.

Practise

Activity 1

Finding the length of an arc

Complete the activity sheet from TES on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper.

Finding the length of an arc

Activity 2

The length of an arc

Complete the activity sheet from White Rose Maths on finding the length of an arc to test your knowledge. You can print it out or write your answers on a piece of paper.

The length of an arc

Click here to see the correct answers.

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