Most students learn a² + b² = c² and immediately start plugging in numbers — without ever understanding why it is true. The visual proof takes two minutes and makes the theorem unforgettable. Then the applications follow naturally.
The Pythagorean Theorem
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Book free demoWhy Is It True? The Area Proof
Draw a square on each side of the right triangle. The area of the square on the hypotenuse c exactly equals the combined area of the squares on legs a and b.
Finding the Hypotenuse (the Longest Side)
When you know both legs and need the hypotenuse, apply the formula directly.
Find c: legs a = 3 and b = 4
Write the formula
Substitute the known values
Take the square root
The 3-4-5 triangle is the most famous Pythagorean triple — a set of whole numbers that satisfy the theorem. Others include 5-12-13 and 8-15-17.
Find c: legs a = 5 and b = 8 (non-perfect square)
Apply the formula
Take the square root (leave in surd form or approximate)
Finding a Missing Leg
When you know the hypotenuse and one leg, rearrange the formula to isolate the unknown leg.
Find a: hypotenuse c = 13, leg b = 5
Start with the formula
Rearrange to isolate a²
Substitute and calculate
Take the square root
Real-World Problem
A ladder 10 m long leans against a wall. Its base is 6 m from the wall. How high up the wall does it reach?
Draw and label the right triangle
Apply the theorem
Solve
Is a Triangle a Right Triangle? The Converse
The theorem also works in reverse: if you know three side lengths, you can check whether a triangle has a right angle.
Is a triangle with sides 6, 8, and 10 a right triangle?
Test with the two shorter sides as a and b, and the longest side as c
Compare
- The hypotenuse is always opposite the right angle — the longest side.
- To find c: use a² + b² = c², then take the square root.
- To find a leg: rearrange to a² = c² − b², then take the square root.
- Always double-check: the hypotenuse must be greater than each individual leg.
- Pythagorean triples (3-4-5, 5-12-13, 8-15-17) appear in many exam questions — recognise them.
Practice Problems
- 1
Find the hypotenuse of a right triangle with legs 6 and 8.
Hint: 6² + 8² = c²
- 2
A right triangle has hypotenuse 17 and one leg 15. Find the other leg.
Hint: a² = 17² − 15²
- 3
Is a triangle with sides 7, 24, and 25 a right triangle?
Hint: Check if 7² + 24² = 25².
- 4
A diagonal fence crosses a rectangular garden that is 9 m by 12 m. How long is the fence?
Hint: The diagonal is the hypotenuse.
- 5
Find the exact length of the diagonal of a square with side length 5 cm.
Hint: Both legs equal 5.
Next in Geometry
The Pythagorean theorem connects directly to distance formula, trigonometry (SOH-CAH-TOA), and circle geometry. If you are studying geometry in Grade 9 or 10, book a demo session and we will map exactly what to cover next.
