Right-Angled Triangles

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Cambridge IGCSE — International Maths (Extended)

Right-Angled Triangles

The Big Idea: Any right-angled triangle is secretly a puzzle with only three pieces of information — two side lengths and one angle — and if you know any two of the three, you can always find the third, using either Pythagoras' theorem (for sides only) or SOHCAHTOA (when an angle is involved).

Quick Summary

Pythagoras' theorem — links the three sides of a right-angled triangle: a² + b² = c². Only works when you have no angles involved, just side lengths.
SOHCAHTOA — links a side, another side, and an angle using sin, cos and tan. This is what you reach for the moment an angle enters the picture.
Angles of elevation & depression — the angle between the horizontal and your line of sight, looking up or down. Usually solved with the tan ratio.
Exact trig values — for special angles (0°, 30°, 45°, 60°, 90°) sin, cos and tan have exact fraction/surd values you're expected to memorise for non-calculator papers.

1. Pythagoras' Theorem

Pythagoras' theorem only ever deals with sides — no angles required (other than knowing there's a right angle somewhere in the triangle). It tells you how the three side lengths of a right-angled triangle are mathematically tied together.

Think of it like this: Imagine the two shorter sides of the triangle as the width and height of a rectangle. If you built actual squares on each of those two sides, and added up their areas, you'd get exactly the area of the square built on the longest side (the hypotenuse). That's the whole theorem — it's about areas of squares, not lengths directly, even though we usually use it to find lengths.

What is the hypotenuse?

The hypotenuse is the longest side of a right-angled triangle. It's always the side opposite the right angle — meaning it's never touching the little square symbol that marks 90°. Spotting the hypotenuse correctly is honestly half the battle; get this wrong and every calculation after it collapses.

B /| / | / | b c / | / | /_____| A a C (right angle at C) c = hypotenuse (opposite the right angle, always the longest side) a and b = the two shorter sides (it doesn't matter which is which)

The Formula a² + b² = c²

In plain English: "square the two shorter sides, add them together, and that equals the square of the hypotenuse."

Finding the hypotenuse (the longest side)

If you're missing the hypotenuse, you're looking for the biggest unknown, which means you're combining two known things into one bigger thing. That's why you add.

Rearranged for c c = √(a² + b²)

Square the two shorter sides → add them → square root the result.

Finding a shorter side

If you already know the hypotenuse (the biggest piece) and one shorter side, you're now trying to find what's "left over" once you strip that shorter side away from the whole. That's why you subtract instead of add.

Rearranged for a shorter side a = √(c² − b²)

Square the hypotenuse and the known shorter side → subtract (bigger minus smaller) → square root the result.

Memory Hook

Finding the hypotenuse → ADD inside the square root.
Finding a shorter side → SUBTRACT inside the square root (always bigger − smaller).

Common Mistake

smaller

Using Pythagoras with other shapes

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Also in the full note

2. SOHCAHTOA (Trigonometric Ratios)
3. Angles of Elevation & Depression
4. Exact Trig Values
What to Memorise
Concepts Checklist
Exam Tips & Common Traps
Labelling the triangle
Finding a missing length — step by step

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Right-Angled Triangles