Last reviewed: 25 October 2026
Pythagoras’ theorem, which junior school learners often meet as the Pythagorean relationship, is one of the most useful results in all of Mathematics. It lets you find a missing side of a right-angled triangle, and it appears in measurement, construction and geometry questions. This guide explains the rule, shows both types of calculation and works through real-life problems.
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The Rule
In a right-angled triangle, the longest side is opposite the right angle and is called the hypotenuse. If the two shorter sides are a and b and the hypotenuse is c, then:
a² + b² = c²
In words: the square of the hypotenuse equals the sum of the squares of the other two sides. The rule only works for right-angled triangles, so check for the small square marking the right angle before using it.
Why It Works
Draw squares on each side of a right-angled triangle with sides 3 cm, 4 cm and 5 cm. The squares have areas 9 cm², 16 cm² and 25 cm², and 9 + 16 = 25. Counting squares on graph paper is a good way for learners to see this before memorising the formula.
Finding the Hypotenuse
Square the two shorter sides, add them, then take the square root.
- Sides 6 cm and 8 cm: 36 + 64 = 100, and √100 = 10 cm.
- Sides 5 m and 12 m: 25 + 144 = 169, and √169 = 13 m.
- Sides 7 cm and 9 cm: 49 + 81 = 130, and √130 ≈ 11.4 cm (to one decimal place).
Finding a Shorter Side
Square the hypotenuse, subtract the square of the known short side, then take the square root.
- Hypotenuse 15 cm, one side 9 cm: 225 − 81 = 144, and √144 = 12 cm.
- Hypotenuse 25 m, one side 7 m: 625 − 49 = 576, and √576 = 24 m.
The most common mistake is adding when you should subtract. A quick check: the hypotenuse must always be the longest side. If your “shorter side” comes out longer than the hypotenuse, you added instead of subtracting.
Pythagorean Triples
Some whole-number sets fit the rule exactly. Learning them saves time: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25. Multiples also work, so 6, 8, 10 and 9, 12, 15 are triples too. If a question gives sides 30 and 40, you can spot straight away that the hypotenuse is 50.
Is the Triangle Right-Angled?
The rule also works backwards. If a² + b² = c² for the three sides, the triangle has a right angle. Sides 9, 40 and 41: 81 + 1,600 = 1,681 = 41², so yes. Sides 6, 7 and 9: 36 + 49 = 85, which is not 81, so no. Builders use this idea with a 3-4-5 rope to check that walls are square.
Real-Life Problems
- Ladder: a 5 m ladder leans against a wall with its foot 3 m from the wall. How high does it reach? 25 − 9 = 16, so 4 m.
- Diagonal of a rectangle: a school field is 80 m by 60 m. The diagonal path is √(6,400 + 3,600) = √10,000 = 100 m.
- Directions: a cyclist rides 12 km north then 9 km east. The straight-line distance from the start is √(144 + 81) = √225 = 15 km.
Always sketch a diagram and label the hypotenuse first. Most errors come from putting the numbers in the wrong places.
Squares and Square Roots
Learners should know the squares of 1 to 20 by heart (for example 13² = 169, 17² = 289) and be comfortable finding square roots, using tables or a calculator where allowed. Strong recall of squares makes these questions much faster.
Practise and Track Progress
Do two questions of each type daily until they feel automatic. The exam countdown offers free daily questions and the What to Revise report points to weak topics. See also our linear equations guide and the Grade 9 Mathematics page. Full timed papers with worked answers are in the Monthly Pass (KSh 500 by M-Pesa).