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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
Students often subtract in the wrong order, or forget which side is the hypotenuse when the triangle is drawn "tilted" or unusually oriented. Always check: is the side you're solving for touching the right-angle mark? If yes, it's a shorter side. If it's the side directly opposite that little square, it's the hypotenuse — no matter how the triangle is rotated on the page.
Also — if your final hypotenuse answer comes out smaller than one of the other sides, you've made an error somewhere. The hypotenuse must always be the biggest number.
Using Pythagoras with other shapes
Pythagoras isn't just for triangles drawn in isolation — it works with any shape you can split into right-angled triangles. A classic example: finding the diagonal of a rectangle. Draw the diagonal, and you've just created two identical right-angled triangles, where the diagonal itself is now the hypotenuse of each.
Practice Question 1
In triangle ABD, AB = 12 cm and AD = 9 cm, with a right angle at D. Find the length of BD.
Practice Question 2
A rectangular field is 40 m long and 30 m wide. Find the length of the diagonal path across it.
2. SOHCAHTOA (Trigonometric Ratios)
The moment a right-angled triangle question involves an angle — either you're given one and need a side, or you're given two sides and need to find an angle — Pythagoras can't help you anymore. That's the signal to switch to SOHCAHTOA.
Think of it like this: SOHCAHTOA is really just three "recipes," each turning two pieces of information (a side and an angle, or two sides) into the third. Which recipe you use depends entirely on which two sides are relevant to your chosen angle.
Labelling the triangle
Before anything else, you must label the three sides relative to your chosen angle, θ (theta):
B
/|
/ |
Hypotenuse / | Opposite
(H) / | (O)
/ |
/_ θ__|
A A C
Adjacent (A)
H = Hypotenuse: always the longest side, always opposite the right angle
O = Opposite: the side directly across from your chosen angle θ
A = Adjacent: the side next to θ (that isn't the hypotenuse)
Key Insight
The hypotenuse never changes — it's always the longest side. But Opposite and Adjacent swap depending on which angle you pick as θ. If you move θ to the other non-right angle in the same triangle, the sides labelled O and A literally swap places. Always relabel from scratch for each new angle.
The Three Ratios
sin θ = O / H cos θ = A / H tan θ = O / A
SOHCAHTOA is the mnemonic: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Finding a missing length — step by step
Label the triangle's sides as H, O, A relative to the given/known angle.
Identify which two letters you're working with (the one you're given and the one you want) — this tells you whether to use sin, cos or tan.
Substitute into the correct formula, remembering brackets around the angle, e.g. sin(50) = A/7.
Rearrange to solve for the unknown — you'll either multiply or divide.
This works almost the same way, except now you have two sides and want the angle. The key extra step is using the inverse trig function (sin⁻¹, cos⁻¹, tan⁻¹) — usually the SHIFT + sin/cos/tan buttons on your calculator.
Example
tan θ = 3/4 → θ = tan⁻¹(3/4)
Round angle answers to 1 decimal place unless told otherwise.
Before You Touch Your Calculator
Make sure it's set to degrees (look for a "D" or "Deg" at the top of the screen). Working in the wrong angle mode (radians or gradians) is one of the most common exam mistakes — and it silently gives you a completely wrong but "plausible-looking" number.
Worked Example — Finding a Length
Find x in a right-angled triangle where the angle is 43°, the adjacent side is 9 cm, and x is the opposite side.
Worked Example — Finding an Angle
Find y° where the adjacent side = 8 cm and the hypotenuse = 23 cm.
Common Mistake
Mislabelling O and A relative to the wrong angle is the #1 error here. A close second: forgetting to use the inverse function when solving for an angle (writing θ = 3/4 instead of θ = tan⁻¹(3/4)) — this gives a nonsensical tiny decimal instead of a real angle.
Shortest distance from a point to a line
This phrase always means the same thing: the perpendicular (right-angle) distance. Whenever you see it, your job is to form a right-angled triangle using that perpendicular line, then apply SOHCAHTOA as normal.
Practice Question 3
A ladder leans against a wall. It makes a 65° angle with the ground, and reaches 4.5 m up the wall. How long is the ladder?
Practice Question 4
A right-angled triangle has a base of 6 cm and height of 8 cm. Find the angle between the base and the hypotenuse.
3. Angles of Elevation & Depression
These are just SOHCAHTOA problems wearing a real-world costume — the maths doesn't change, but the setup requires you to correctly interpret a description of someone looking up or down at something.
Angle of Elevation
The angle between the horizontal and your line of sight when looking UP at something (e.g. looking up at a bird, a plane, or the top of a building).
Angle of Depression
The angle between the horizontal and your line of sight when looking DOWN at something (e.g. looking down from a cliff at a boat).
Crucial Rule: Alternate Angles
If a person at the top of a cliff looks down at a boat with an angle of depression of 35°, then — because the horizontal lines (at the person's eye level and at the boat's level) are parallel — the angle of elevation from the boat looking up at the person is also 35°. These are alternate angles between parallel lines. This trick is used constantly in these problems to transfer the angle into the triangle where you actually need it.
Object Angle of
\ elevation
\ ↗
\ /
\ / ← Line of sight
─────────\──/────────── Horizontal
\/
\
\ ← Line of sight
\
\ ↙ Angle of
\ depression
Boat
Because these scenarios often involve two overlapping right-angled triangles (like a cliff, a boat, and a flag marker partway up the cliff), it's worth sketching each triangle separately once you've drawn the full picture — trying to solve everything from one messy combined diagram is where mistakes creep in.
Worked Example
A cliff top T stands 24 m above sea level. The angle of depression from T to a boat B is 35°. Find the distance from the boat to the foot of the cliff, F.
Practice Question 5
From the top of a lighthouse 40 m tall, the angle of depression to a ship is 22°. How far is the ship from the base of the lighthouse?
4. Exact Trig Values
For a handful of special angles, sin, cos and tan produce clean fractions and surds instead of messy decimals. These come up specifically in non-calculator questions, so memorising them is essential — you can't just tap them into a calculator on the day.
θ
0°
30°
45°
60°
90°
sin θ
0
1/2
√2/2
√3/2
1
cos θ
1
√3/2
√2/2
1/2
0
tan θ
0
√3/3
1
√3
undefined
Memory Trick
Remember sin θ using the pattern: √0/2, √1/2, √2/2, √3/2, √4/2 — which simplifies to 0, 1/2, √2/2, √3/2, 1.
Then notice: cos θ is just sin θ reversed — the values from 0° to 90° for sin match the values from 90° to 0° for cos.
Two triangles that generate all of these
The 45° Triangle
Take an isosceles right-angled triangle with both shorter sides = 1. By Pythagoras, the hypotenuse = √(1² + 1²) = √2.
45°
/|
√2/ | 1
/ |
/___|
1
sin 45° = 1/√2 = √2/2
cos 45° = 1/√2 = √2/2
tan 45° = 1/1 = 1
The 30°/60° Triangle
Take an equilateral triangle with all sides = 2, and slice it exactly in half. The base splits into two pieces of 1, and by Pythagoras the height = √(2² − 1²) = √3.
/|\
/ | \
2 / | \ 2
/ |√3 \
/30° | \
/_____|_____\
1 1
sin 30° = 1/2 sin 60° = √3/2
cos 30° = √3/2 cos 60° = 1/2
tan 30° = 1/√3 = √3/3 tan 60° = √3
Exam Tip
Sketch both of these triangles at the very start of your non-calculator exam, before you even read the questions. It costs you 30 seconds and means you can just glance back at them any time you need an exact value, instead of trying to recall the table from memory under pressure.
Using exact values in calculations
Simply substitute the exact value in and solve as normal — treat it exactly like any other number, surds and all.
Practice Question 6
In a right-angled triangle, angle A = 60° and the hypotenuse = 28 cm. Find the side opposite angle A, in exact (surd) form.
Practice Question 7
The point (30, k) lies on the graph y = tan x. Find the exact value of k.
What to Memorise
Pythagoras' Theorem
a² + b² = c² — c is always the hypotenuse (longest side, opposite the right angle).
Finding hypotenuse
c = √(a² + b²) — ADD inside the root.
Finding a shorter side
a = √(c² − b²) — SUBTRACT inside the root (bigger minus smaller).
SOHCAHTOA
sin θ = O/H, cos θ = A/H, tan θ = O/A
Hypotenuse
The longest side of a right-angled triangle; always opposite the right angle.
Angle of elevation
Angle above the horizontal when looking UP at something.
Angle of depression
Angle below the horizontal when looking DOWN at something.
Alternate angles rule
Angle of depression from the top = Angle of elevation from the bottom (parallel horizontals).
Exact Trig Values Table (memorise this cold)
θ
0°
30°
45°
60°
90°
sin θ
0
1/2
√2/2
√3/2
1
cos θ
1
√3/2
√2/2
1/2
0
tan θ
0
√3/3
1
√3
undefined
Concepts Checklist
Exam Tips & Common Traps
Trap 1: Wrong hypotenuse
If your final "hypotenuse" answer is smaller than one of the other sides, something went wrong. Always sanity-check: the hypotenuse must be the biggest number in the triangle.
Trap 2: Add vs subtract
Adding when you should subtract (or vice versa) inside the square root is the single most common Pythagoras error. Finding the hypotenuse = ADD. Finding a shorter side = SUBTRACT (bigger − smaller).
Trap 3: Mislabelling O and A
O and A depend on which angle you pick as θ. Relabel fresh every time you switch which angle you're working from — don't assume the labelling carries over.
Trap 4: Calculator in the wrong mode
Degrees, not radians! Check for "D" or "Deg" on your calculator display before starting any trig question.
Trap 5: Rounding too early
In multi-step questions, keep exact values (like √63) in your working and only round at the very final answer. Rounding early compounds errors and can cost accuracy marks even if your method is correct.
Trap 6: Forgetting the inverse function
When solving for an angle, you must use sin⁻¹, cos⁻¹, or tan⁻¹ — never just leave it as "θ = 3/4."
What examiners are actually looking for
Clear identification of which side/angle is which (H, O, A) shown in your working — method marks are often awarded even if the final answer is wrong.
Correct use of brackets around angles when substituting, e.g. sin(50), not sin 50 written ambiguously.
Answers rounded to the requested degree of accuracy (commonly 3 s.f. for lengths, 1 d.p. for angles) — unless the question asks for an exact answer.
For multi-step problems, a diagram showing each separate right-angled triangle you're using, especially in elevation/depression problems with overlapping triangles.
Recognition of alternate angles when transferring an angle of depression into a triangle where it becomes useful as an angle of elevation.