Right-angled triangle

Properties

• One angle is 90° and the other two add up to 90°.
• The hypotenuse is the longest side.
• The sides obey Pythagoras: the square on the hypotenuse equals the sum of the squares on the legs.
• Whole-number sides such as 3, 4, 5 and 5, 12, 13 are called Pythagorean triples.

Why the formulas work

The two legs meet at a right angle, so one leg is the base and the other is the height: the area is half their product, with no extra measuring. The hypotenuse comes from the theorem of Pythagoras, c² = a² + b². You can check it with a 3, 4, 5 triangle: 9 + 16 = 25.

Worked example

Take a right-angled triangle with leg a 6 cm and leg b 8 cm.

1. Hypotenuse. Hypotenuse c = √(a² + b²). Here: √(6² + 8²) = 10 cm.
2. Area. Area = ½ × a × b. Here: ½ × 6 × 8 = 24 cm².
3. Perimeter. Perimeter = a + b + c. Here: 6 + 8 + 10 = 24 cm.

Finding a missing measurement

If the hypotenuse and one leg are known, subtract the squares. A hypotenuse of 17 cm and a leg of 8 cm give the other leg as √(289 − 64) = √225 = 15 cm. The area is then ½ × 8 × 15 = 60 cm².

Mistakes to avoid

• Adding the legs and taking the root: √(a + b) is wrong. Square first, add, then take the root.
• Using the hypotenuse as the height when finding the area.
• Pythagoras works only in right-angled triangles.

Where you see it

A ladder against a wall, a ramp, the diagonal of a screen, and the path across a rectangular park corner to corner all make right-angled triangles. Masons use a 3, 4, 5 rope to set out a true right angle.

Try these

Work each one out, then tap to check.

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