The Pythagorean Theorem

In a right triangle, the two shorter sides squared and added together equal the longest side squared.

a² + b² = c²

The two legs a and b meet at the right angle. The hypotenuse c is always opposite it.

What each symbol means

The Pythagorean theorem only works for right triangles, which are triangles with one square corner (a 90 degree angle). For those triangles it links the three sides with one neat rule.

The rule is a² + b² = c². Square the two short sides, add them, and you get the square of the longest side. The longest side, called the hypotenuse, is always the one across from the right angle.

Which side is c?: The hypotenuse (c) is always the longest side and always sits opposite the right angle. The two sides that form the right angle are the legs, a and b.

A worked example

A right triangle has legs 3 and 4. Find the hypotenuse.

  1. Write the rule: a² + b² = c².
  2. Put in the legs: 3² + 4² = c².
  3. Square each: 9 + 16 = c².
  4. Add: 25 = c².
  5. Take the square root of both sides: c = √(25) = 5.
  6. The hypotenuse is 5.

Finding a leg instead: If you know the hypotenuse and one leg, subtract instead of add. For example c² - a² = b², then take the square root.

Finding a shorter side

The hypotenuse is 13 and one leg is 5. Find the other leg.

  1. Rearrange: b² = c² - a².
  2. Put in the numbers: b² = 13² - 5².
  3. Square each: b² = 169 - 25.
  4. Subtract: b² = 144.
  5. Square root: b = √(144) = 12.

Why it works (the four-triangle picture)

  1. Draw a big square and place four identical right triangles inside it, one in each corner, so their hypotenuses form a tilted square in the middle.
  2. The tilted middle square has area c² (its side is the hypotenuse c).
  3. Now slide the same four triangles into a different arrangement inside the same big square. This time they leave two smaller squares of area a² and b².
  4. The big square did not change size, and the four triangles are the same, so the leftover space must match: c² = a² + b².

Practice questions

Question 1

A right triangle has legs 6 and 8. Find the hypotenuse.

  1. To find the hypotenuse, use a² + b² = c².
  2. Put in the legs: 6² + 8² = c².
  3. Square each and add: 36 + 64 = 100, so c² = 100.
  4. Take the square root of both sides: c = √(100) = 10.

Answer: 10

Question 2

A right triangle has legs 5 and 12. Find the hypotenuse.

  1. To find the hypotenuse, use a² + b² = c².
  2. Put in the legs: 5² + 12² = c².
  3. Square each and add: 25 + 144 = 169, so c² = 169.
  4. Take the square root of both sides: c = √(169) = 13.

Answer: 13

Question 3

The hypotenuse is 10 and one leg is 6. Find the other leg.

  1. We know the hypotenuse and one leg, so rearrange to b² = c² - a².
  2. Put in the numbers: b² = 10² - 6².
  3. Square each and subtract: 100 - 36 = 64, so b² = 64.
  4. Take the square root: b = √(64) = 8.

Answer: 8

Question 4

A ladder 5 m long leans so its base is 3 m from the wall. How high up the wall does it reach?

  1. The ladder is the hypotenuse (5 m) and the gap along the ground is one leg (3 m); the height up the wall is the other leg.
  2. Rearrange to find the missing leg: b² = c² - a².
  3. Put in the numbers: b² = 5² - 3² = 25 - 9 = 16.
  4. Take the square root: b = √(16) = 4.
  5. The ladder reaches 4 m up the wall.

Answer: 4 m

Question 5

Are sides 8, 15, 17 a right triangle?

  1. A triangle is right-angled only if the two shorter sides squared add up to the longest side squared, so test a² + b² = c².
  2. Here 17 is the longest side, so check 8² + 15² against 17².
  3. Work out each part: 8² + 15² = 64 + 225 = 289, and 17² = 289.
  4. The two sides match exactly, so the sides do form a right triangle.

Answer: Yes