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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

GCSE Subjects:   Maths

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

Curriculum topic:   Basic Geometry, Geometry and Measures

Curriculum subtopic:   Circles Properties and Constructions

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Musician silhouette


Do you love music? Or maybe musicals?

They really strike a chord.


Music and maths are more closely related than you may think...

Chords appear in maths too.

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


They are used in circle theorems.

Lets see how...


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    

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




e.g. What is the value of angle x? 


Diagram showing chords and radii with a missing angle x

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°




e.g. What is the length of side x? 


Diagram showing chords and radii with a missing side x

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.

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