Loading please wait

The smart way to improve grades

Comprehensive & curriculum aligned

Try an activity or get started for free

Apply Chords and Radii

In this worksheet, students will use the fact that the intersection of a chord and a radius will create a right angle to find missing angles or side lengths.

'Apply Chords and Radii' worksheet

Key stage:  KS 4

Year:  GCSE

GCSE Subjects:   Maths

GCSE Boards:   OCR, AQA, Eduqas, Pearson Edexcel,

Curriculum topic:   Basic Geometry, Geometry and Measures

Curriculum subtopic:   Circles Properties and Constructions

Popular topics:   Geometry worksheets

Difficulty level:  

Worksheet Overview

What is a chord?


A chord is the line segment joining two points on a curve.




You can think of it this way: a chord is a straight line that touches the circumference twice. A chord is different to a diameter because it doesn't pass through the centre of the circle.



Chords in circle theorems


If a radius of a circle bisects a chord, they will create a right angle and the lengths to either side will be equal:


Diagram showing chords and radii    


In this circle, A and B are points on the circumference. O is the centre and M is the point where the radius and chord intersect.

Length AM = BM.

Angle AMO = BMO = 90º.

These facts can be combined with other theorem to find missing angles or sides within a circle - let's explore how in some examples. 



Example 1


e.g. What is the value of angle x? 


Diagram showing chords and radii with a missing angle x. Angle at centre = 48


The rule to use here is: where the radius bisects the chord, a right angle will be created. 

We also know that there is always 180° in total within a triangle


So to find angle x, we can subtract the two known angles from 180° to find x:

180 - 90 - 48 = 42°




Example 2


Calculate length x.


Diagram showing chords and radii with a missing side x. Known lengths = 8 and 6.


As the chord cuts the radius at a 90° angle, we know that a right-angled triangle has been formed.


So, we can apply Pythagoras' Theorem to find our missing side length (x):

a² + b² = c²

6² + x² = 8²

8² - 6² = x² 

√ 28 = 5.29 cm




In this activity, we will use the facts detailed above to find missing angles or side lengths.


As with all circle theorems, we will need to recall other angle and shape properties as well, so watch out for these.

What is EdPlace?

We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.

Get started

Popular Maths topics

Try an activity or get started for free