How it works
In a right triangle, the two sides that form the right angle are the legs (call them a and b), and the side across from the right angle is the hypotenuse c. The hypotenuse is always the longest side. The Pythagorean theorem says
a² + b² = c²
To find the hypotenuse, add the squares of the legs and take the square root: c = √(a² + b²). To find a leg, subtract instead: a = √(c² − b²). A common mistake is adding when you should subtract, so always ask first: am I looking for the longest side or not?
Pythagorean triples save time
Whole-number triples come up constantly in contests: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29 and 9-40-41, plus their multiples such as 6-8-10 and 9-12-15. If you recognise one, you can skip the square root entirely.
Look for the hidden right triangle
Many problems never say "right triangle". The diagonals of a rhombus cross at right angles, the height of an isosceles triangle splits it into two right triangles, and the distance between two points on a grid is the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps.
Pythagorean Theorem practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
A right triangle has legs 3 and 4. Find the hypotenuse.
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Answer: 5
c² = 3² + 4² = 9 + 16 = 25, so c = 5. This is the 3-4-5 triple.
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
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Answer: 12
We want a leg, so subtract: 13² − 5² = 169 − 25 = 144, and √144 = 12. (5-12-13 is a triple.)
A rhombus has diagonals 10 and 24. Find its perimeter.
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Answer: 52
The diagonals of a rhombus bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 10 ÷ 2 = 5 and 24 ÷ 2 = 12. Each side is √(25 + 144) = 13, so the perimeter is 4 × 13 = 52.
An isosceles triangle has base 24 and equal sides 20. Find its area.
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Answer: 192
The height from the top vertex splits the base into two halves of 12. The height is √(20² − 12²) = √(400 − 144) = √256 = 16. Area = ½ × 24 × 16 = 192.
A right triangle has legs that sum to 17 and hypotenuse 13. Find its area.
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Answer: 30
Let the legs be a and b. Then (a + b)² = a² + 2ab + b², so 289 = 169 + 2ab, giving ab = 60. The area is ab ÷ 2 = 30. (No need to find each leg.)
Keep going
Related topics
- Special right triangles: 45-45-90 and 30-60-90
- Triangle area
- Triangle inequality
- Circle area and circumference
Related practice sets in the app
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice