Maths · KS3 and GCSE · Interactive lesson
Right-angled trigonometry: choosing a ratio
Name the opposite, adjacent and hypotenuse sides. Use the marked angle to choose sine, cosine or tangent in a right-angled triangle.
Choose your view
Explore an interactive example, then try the questions at your own pace.
Later KS3 / GCSE
By the end of this lesson: Name the opposite, adjacent and hypotenuse for a chosen angle in a right-angled triangle.
The interactive diagram and controls are in Stage 2, “Explore one example”. Read the idea here, then try it next.
- A right-angled triangle has one 90° angle.
- The hypotenuse is across from the right angle. It is the longest side.
- Choose an angle smaller than 90°: opposite is across from it; adjacent touches it but is not the hypotenuse.
Useful starting points
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What do the words mean?
- Acute angle
- An angle smaller than 90°.
- θ (theta)
- The marked angle. This letter is just a name for the angle.
- Hypotenuse
- The side opposite 90 degrees.
- Adjacent
- The side beside the marked angle, excluding the hypotenuse.
- Opposite
- The side across from the marked angle.
Change the example
This is the lesson example. Your practice question uses its own numbers.
Something to try
Increase the marked angle. Watch which side is opposite it. The hypotenuse stays opposite the right angle.
Notice what changes
How particle motion works +
Starts paused. Press Play, then add energy to see the warmer particles move faster.
Particles keep the same size. Liquid particles move around; solid particles vibrate around fixed positions. This is a simplified model, with the change in motion exaggerated to make it easier to see.
About the playback +
About this model
The marked angle is at the left. These side names refer to that angle. The little square marks 90°.
More about this diagram
Still picture. Play particle motion in Stage 2.
Name the sides before doing a calculation.
Still picture of the example from Stage 2. Use the back arrow to change it.
Work through it
Starting example: read the steps below
- Find the little square: that is the right angle.
- Across from the square is the hypotenuse.
- Across from the marked angle is the opposite side.
- Beside the marked angle, excluding the hypotenuse, is the adjacent side.
Try on your own
Your turn. No rush.
Question 1
Which side is always opposite the right angle?
About these questions
Questions appear in a mixed order. Work at your own pace and stop whenever you like.
These questions use their own numbers, separate from the lesson example. You can stop whenever you like. Answers last for this visit to the page.
A good place to pause
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Can you name the sides from the marked angle before choosing a ratio?
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Notes for parents and tutors
These lessons cover selected KS3 and GCSE ideas for England; they are not a complete course. The level notes explain where GCSE methods or optional extensions begin. Maurice can help you choose material for your course and exam board.
The content draws on the DfE Maths curriculum, DfE Science curriculum and AQA GCSE Physics specification. Level labels are guidance, not exam-board endorsements. For AQA Radioactivity, calculating net decline as a ratio is Higher-tier work. Net means the overall change; ask Maurice whether this extension is relevant to your course. Explanations and practice were reviewed during development on 21 October 2026; Maurice should check their fit before assigning work.
Topic labels and your lesson-view preference are saved in this browser. This tab also remembers the last lesson stage, so you can choose to continue there. If you choose revision, selected activities and suggested revisit dates are saved here too. Recent question codes are also kept to reduce repeats. Reflection notes are saved automatically per topic when you type them; you can delete a note in the recap. They stay in this browser and are visible to other people using it. Practice answers and enquiry details are not saved. Clearing browser data removes the labels.