Pythagoras' theorem step by step
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c². Here a and b are the legs, the two shorter sides next to the right angle. c is the hypotenuse, the longest side. We square, add or subtract, and then take the square root.
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Legs and hypotenuse
The theorem only holds in a right-angled triangle, where one corner is exactly 90 degrees.
- The legs a and b: the two shorter sides, which form the right angle.
- The hypotenuse c: the side opposite the right angle. It is always the longest.
a² + b² = c²
a² is the area of a square with side a. Together, the squares on the two legs have the same area as the square on the hypotenuse.
The hypotenuse or a leg?
| We want to find | We calculate |
|---|---|
| The hypotenuse c | c² = a² + b², then c = √(c²) |
| A leg a | a² = c² − b², then a = √(a²) |
First we get an area, for example c² = 225. We get the side with the square root: √225 = 15, because 15 · 15 = 225.
Is the triangle right-angled?
We square the two shortest sides and add them. If the sum equals the square of the longest side, the triangle is right-angled.
The sides are 7, 9 and 12 cm: 7² + 9² = 49 + 81 = 130 and 12² = 144.
130 ≠ 144
The numbers are different, so the triangle is not right-angled.
Example
Example 1: the hypotenuse when the legs are 9 cm and 12 cm
We want to find the hypotenuse c. Both legs are known, so we add their squares.
We put the legs into the theorem.
9² + 12² = c²
We work out the squares: 9 · 9 = 81 and 12 · 12 = 144.
81 + 144 = c²
We add. This is the area of the square on the hypotenuse.
c² = 225
We want the side, not the area, so we take the square root.
c = √225 = 15
The hypotenuse is 15 cm.
15 cm is longer than both legs, as a hypotenuse should be. And 15² = 225 = 81 + 144.
Example 2: a leg when the hypotenuse is 25 cm and one leg is 7 cm
We want to find the leg a. We take the known small square away from the large one.
The hypotenuse, 25, goes on its own in the place of c.
a² + 7² = 25²
We work out the squares: 7 · 7 = 49 and 25 · 25 = 625.
a² + 49 = 625
We want a² on its own. We subtract 49 from both sides, so the two sides stay equal.
a² + 49 − 49 = 625 − 49
On the left, 49 − 49 is zero. On the right, we get 576.
a² = 576
We take the square root.
a = √576 = 24
The leg is 24 cm.
24 cm is shorter than the hypotenuse, 25 cm. And 24² + 7² = 576 + 49 = 625 = 25².
Pitfalls
Adding the sides without squaring: 9 + 12 = 21.
The theorem is about squares: 9² + 12² = 81 + 144 = 225, and √225 = 15.
Putting the hypotenuse in as a or b and adding when we want a leg.
The side opposite the right angle is c and goes on its own. For a leg, a² = c² − b².
Forgetting the square root and answering c = 225 instead of c = 15.
c² is an area. We get the side with the square root.
Practise on your own
0 of 4 correct1A right-angled triangle has legs of 5 cm and 12 cm. How long is the hypotenuse?
2The hypotenuse of a right-angled triangle is 10 cm and one leg is 6 cm. How long is the other leg?
3A triangle has sides of 8 cm, 15 cm and 17 cm. Is it right-angled? Answer yes or no.
4A right-angled triangle has legs of 3 cm and 7 cm. How long is the hypotenuse? Round to one decimal place.
Common questions
Does Pythagoras' theorem hold in all triangles?
No, only in right-angled triangles. In other triangles, a² + b² is greater or smaller than c².
How do you work out the diagonal of a TV or a rectangle?
The diagonal is the hypotenuse of a right-angled triangle. A screen that is 80 cm wide and 45 cm high has a diagonal of √(80² + 45²) = √8425 ≈ 91.8 cm.
Why is it called Pythagoras' theorem?
It is named after the Greek mathematician Pythagoras, who lived about 2 500 years ago. The relationship was known even earlier, for example in Babylonia.
Read also
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- Years 4–9Angles: types, angle sum, angles on a lineHow do you work out a missing angle?
- Years 6–9Solving equations step by stepHow do you solve an equation?
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