Loading please wait

The smart way to improve grades

Comprehensive & curriculum aligned

Try an activity or get started for free

Find Side Lengths on a Triangle Using Trigonometry

In this worksheet, students will use trigonometry to calculate a side length in right-angled triangles.

'Find Side Lengths on a Triangle Using Trigonometry ' worksheet

Key stage:  KS 3

Year:  Year 9 Maths worksheets

Curriculum topic:   Geometry and Measures

Curriculum subtopic:   Solve Right-Angled Triangle Problems

Popular topics:   Angles worksheets, Geometry worksheets

Difficulty level:  

Worksheet Overview

In a right-angled triangle, the sides and the angles are connected by three trigonometric ratios.



The three trig ratios are:



The best way to use these is in a formula triangle:


For example with sin:


sine formula triangle


The triangle above shows the formula triangle for using sin. 

To use it, we simply cover up the part of the triangle that we want to find out and then rearrange the equation.


For example, if we know the sin and the opposite side, we cover up H (the hypotenuse) and we can see that we need to do: O ÷ S = H see below


sine formula triangle


If we know the sin and the hypotenuse, we cover up O and we can do this equation: S x H = O see below:


sine formula triangle


Clearly, if we know O and A but don't know S, we'd cover up S and do O ÷ H = S


We can remember the triangle by using the letters SOH to spell a sort of word!


We can do the same for the other two formulae too:


Here is the triangle for cos - we can remember this one by the word CAH


cosine triangle


And here's the one for tan - we can remember this one by the word TOA


tangent triangle


We can stick the words together to make a pretty silly-looking word:





Using trigonometry, calculate the side length x to 3 sig. figs.




Label the sides adjacent, opposite and hypotenuse in relation to the given angle 34º.

The given sides are the opposite (x) and the hypotenuse (8). We don't need to know the adjacent side.


So, we know the hypotenuse and the angle, and we need to find out the opposite side. So which formula do we need that has S, O and H in it?

It's the SOH one with SIN!



Which letter do we need to cover up in the triangle? The side we want to find out is O, so cover up O and we do H x Sin


x = 8 sin34º

x = 8 x  0.559192903....

x = 4.47 (3 s.f.)


It looks very complicated but really it's very simple once you've chosen the correct formula triangle to use!


Let's practise this in the questions that follow.

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

Try an activity or get started for free