The Pythagorean Theorem, Explained
In any right triangle, the square on the longest side equals the sum of the squares on the other two. It only works for right triangles, and the longest side is always the one opposite the right angle.
What the letters mean
- c is the hypotenuse: the side opposite the right angle, always the longest.
- a and b are the legs: the two sides that form the right angle.
- The relationship is a^2 + b^2 = c^2, and c is never one of the legs.
Finding a leg instead
Rearrange rather than starting over. If you know c and one leg, then a^2 = c^2 - b^2. The subtraction is the whole difference, and doing it in the wrong order is what produces a negative under the root.
Triples worth recognising
- 3, 4, 5 and its multiples: 6-8-10, 9-12-15, 30-40-50.
- 5, 12, 13.
- 7, 24, 25.
- 8, 15, 17.
Worked examples
Example 1
Question
A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
Answer
10 cm
Steps
- 1Both given sides form the right angle, so both are legs.
- 2Apply the theorem: c^2 = 6^2 + 8^2 = 36 + 64 = 100.
- 3Take the positive square root: c = 10 cm.
- 4Recognise the pattern: this is the 3-4-5 triple doubled.
Example 2
Question
A right triangle has a hypotenuse of 13 m and one leg of 5 m. Find the other leg.
Answer
12 m
Steps
- 1The 13 is the hypotenuse, so it is c, not a leg.
- 2Rearrange: b^2 = c^2 - a^2 = 13^2 - 5^2.
- 3That is 169 - 25 = 144.
- 4Take the root: b = 12 m.
- 5Check: 5^2 + 12^2 = 25 + 144 = 169, which is 13^2.
Adding instead of subtracting here gives root 194, about 13.9, which is longer than the hypotenuse. A leg longer than the hypotenuse is impossible, and noticing that catches the error.
Example 3
Question
A 5 m ladder leans against a wall with its base 1.4 m from the wall. How high does it reach?
Answer
4.8 m
Steps
- 1The ladder is the hypotenuse because it is opposite the right angle between wall and ground.
- 2So height^2 = 5^2 - 1.4^2 = 25 - 1.96.
- 3That gives 23.04.
- 4Take the root: height = 4.8 m.
- 5Sense check: the height is less than the ladder length, as it must be.
Common mistakes
Treating a given side as a leg when it is actually the hypotenuse.
Find the right angle first. The side opposite it is c, and it is always the longest.
Using the theorem on a triangle with no right angle.
It only applies to right triangles. Without one, you need the cosine rule instead.
Forgetting to take the square root at the end and reporting 100 instead of 10.
c^2 = 100 means c = 10. The last line is the root, not the square.
Adding the squares when you were asked for a leg.
Finding a leg is subtraction: c^2 - a^2. If the answer comes out longer than the hypotenuse, you added.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. Legs of 9 and 12. Find the hypotenuse.
Show answer
15. 81 + 144 = 225, and the root of 225 is 15.
2. Hypotenuse 17, one leg 8. Find the other leg.
Show answer
15. 289 - 64 = 225, and the root is 15.
3. Is a triangle with sides 7, 24, and 25 a right triangle?
Show answer
Yes. 49 + 576 = 625, which equals 25^2, so the theorem holds and the angle is right.
Questions people ask
- Does it work for any triangle?
- No, only right triangles. For others, use the cosine rule, which reduces to Pythagoras when the angle is 90 degrees.
- How do I know which side is the hypotenuse?
- It is opposite the right angle. It is also always the longest side, which is a quick sanity check on any answer.
- Can I use it to test whether an angle is right?
- Yes. If a^2 + b^2 equals c^2 the angle is right, if the sum is bigger the angle is acute, and if smaller it is obtuse.
- Does it work in three dimensions?
- Yes, applied twice. Find the diagonal of the base first, then treat that as a leg with the height to reach the space diagonal.
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