What you'll learn
Pythagoras' theorem is one of the most useful results in the CSEC Mathematics syllabus, sitting within Geometry and Trigonometry. It connects the three sides of any right-angled triangle, allowing you to find a missing length whenever a right angle is present. In this guide you will learn the theorem itself, how to find the hypotenuse and how to find a shorter side, how to apply it to real situations such as ladders, ramps and distances, and how to extend it to three-dimensional problems like the diagonal of a box. You will also learn to recognise Pythagorean triples for quick checks. The theorem appears in Paper 1 and Paper 2 and is a building block for coordinate geometry, trigonometry and the cosine rule.
Key terms and definitions
Right-angled triangle — a triangle containing one 90° angle.
Hypotenuse — the longest side of a right-angled triangle, always opposite the right angle.
Pythagoras' theorem — for a right-angled triangle, c² = a² + b², where c is the hypotenuse.
Pythagorean triple — a set of three whole numbers that fit the theorem exactly, e.g. 3, 4, 5.
Diagonal — a straight line joining two non-adjacent corners, often the unknown in 2D and 3D problems.
Converse — the reverse statement: if c² = a² + b², the triangle is right-angled.
Core concepts
The theorem
In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
c² = a² + b².
Here c is the hypotenuse (opposite the right angle) and a and b are the two shorter sides (the legs). The theorem only works when there is a right angle, so always confirm or look for the 90° mark.
Finding the hypotenuse
When the two shorter sides are known, add their squares and take the square root. If a = 6 and b = 8, then c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10. The hypotenuse is always the largest side, so your answer for c should be bigger than either leg — a quick sanity check.
Finding a shorter side
When the hypotenuse and one leg are known, subtract: a² = c² − b². If the hypotenuse is 13 and one leg is 5, then a² = 13² − 5² = 169 − 25 = 144, so a = 12. The most common error here is adding instead of subtracting; remember that a shorter side must come out smaller than the hypotenuse.
Pythagorean triples
Some side lengths are whole numbers that satisfy the theorem exactly: 3-4-5, 5-12-13, 8-15-17, and their multiples (such as 6-8-10). Recognising these saves time and helps you check answers, but always be ready to work with surds and decimals for non-triple triangles.
The converse (testing for a right angle)
If the three sides of a triangle satisfy c² = a² + b² (with c the longest), then the triangle must be right-angled. This converse is useful for checking whether a corner is square — a technique builders use with the 3-4-5 method.
Three-dimensional problems
In 3D, Pythagoras is applied twice. To find the longest diagonal of a cuboid with edges a, b and c, first find the diagonal of the base (√(a² + b²)), then combine it with the height: the space diagonal is √(a² + b² + c²). Drawing the right-angled triangle inside the solid is the key skill.
Worked examples
Example 1: Finding the hypotenuse (Paper 1/2 style)
A ramp rises 2.5 m vertically over a horizontal distance of 6 m. Find the length of the ramp surface.
The ramp is the hypotenuse of a right-angled triangle with legs 2.5 m and 6 m. So c² = 2.5² + 6² = 6.25 + 36 = 42.25, giving c = √42.25 = 6.5 m.
Example 2: Finding a shorter side (Paper 2 style)
A 5 m ladder leans against a wall with its foot 1.4 m from the base. How far up the wall does it reach?
The ladder is the hypotenuse (5 m), the distance from the wall is one leg (1.4 m), and the height up the wall is the unknown leg h. So h² = 5² − 1.4² = 25 − 1.96 = 23.04, giving h = √23.04 = 4.8 m.
Example 3: A 3D diagonal (Paper 2 style)
A storage box measures 8 cm by 6 cm by 24 cm. Find the length of the longest straight rod that fits inside (the space diagonal).
The space diagonal is √(8² + 6² + 24²) = √(64 + 36 + 576) = √676 = 26 cm.
Common mistakes and how to avoid them
Adding when you should subtract. Use c² = a² + b² only to find the hypotenuse. To find a shorter side, subtract: a² = c² − b².
Misidentifying the hypotenuse. The hypotenuse is always opposite the right angle and is the longest side; never label a leg as the hypotenuse.
Forgetting to take the square root. After finding c² = 100, the side is √100 = 10, not 100.
Using it without a right angle. Pythagoras applies only to right-angled triangles. If there is no right angle, use the cosine rule instead.
Rounding too early in multi-step or 3D problems. Keep full accuracy until the final answer.
Exam technique for Pythagoras' theorem
Sketch and label. Mark the right angle and identify which side is the hypotenuse before writing the equation.
Decide add or subtract first. Finding the longest side → add; finding a shorter side → subtract. State c² = a² + b² (or the rearrangement) clearly.
Sanity-check the size. The hypotenuse must be the biggest; a leg must be smaller than the hypotenuse.
For 3D, build a right-angled triangle. Use the base diagonal first, then the height, or apply √(a² + b² + c²) directly for a cuboid.
Round sensibly. Give the final answer to the accuracy requested, usually 1–3 significant figures, and include units.
Quick revision summary
Pythagoras' theorem, c² = a² + b², links the sides of a right-angled triangle, where c is the hypotenuse opposite the right angle. To find the hypotenuse, add the squares of the two legs and take the square root; to find a shorter side, subtract (a² = c² − b²). The hypotenuse is always the longest side, which gives a quick check on your answer. Recognise common triples (3-4-5, 5-12-13, 8-15-17) to save time, and use the converse to test whether a triangle is right-angled. In 3D, apply the theorem twice or use √(a² + b² + c²) for the space diagonal of a cuboid, always by drawing the relevant right-angled triangle. The theorem applies only where a right angle exists — otherwise use the cosine rule. Sketch and label first, decide whether to add or subtract, keep full accuracy until the end, and include units in your final answer.