Pythagorean Theorem (a² + b² = c²)

First, meet the right triangle

Every right triangle has two short sides called legs (we call them a and b) and one long side called the hypotenuse (we call it c). The hypotenuse is always the side opposite the right angle, the longest side, and the one that never touches the square corner.

Legs a and b form the right angle; the hypotenuse c sits across from it.

The rule: a² + b² = c²

The theorem says that if you square both legs and add them together, you get the square of the hypotenuse. In symbols:

a² + b² = c²

a, b  =  the two legs (the short sides)
c     =  the hypotenuse (longest side, across from the 90° angle)

• Squaring a number just means multiplying it by itself: 5² = 5 × 5 = 25.
• To find the hypotenuse (c), add the two squared legs, then take the square root.
• To find a missing leg (a or b), subtract instead: rearrange to a² = c² − b², then take the square root.

Worked example #1: find the hypotenuse (legs 3 and 4)

A right triangle has legs of 3 cm and 4 cm. How long is the hypotenuse? Since we want c, we square both legs, add, and take the square root.

• Square the legs: 3² = 9 and 4² = 16.
• Add them: 9 + 16 = 25, so c² = 25.
• Square root: c = √25 = 5.

  a² + b² = c²
  3² + 4² = c²      (put in the legs)
   9 + 16 = c²      (square each leg)
       25 = c²
        c = √25 = 5  (square root both sides)

Worked example #2: find a missing leg (the ladder problem)

A 13-foot ladder leans against a wall. The base of the ladder sits 5 feet out from the wall. How high up the wall does the ladder reach? The ladder is the hypotenuse (c = 13), the distance along the ground is one leg (a = 5), and the height up the wall is the missing leg (b).

• Rearrange for the leg: b² = c² − a².
• Square the known sides: 13² = 169 and 5² = 25.
• Subtract: 169 − 25 = 144, so b² = 144.
• Square root: b = √144 = 12.

  a² + b² = c²
  5² + b² = 13²      (ladder = hypotenuse = 13)
  25 + b² = 169
       b² = 169 − 25  (subtract to isolate the leg)
       b² = 144
        b = √144 = 12

Worked example #3: diagonal distance across a park

A rectangular park is 8 m wide and 6 m deep. Instead of walking around two sides, you cut straight across the diagonal. How far is that diagonal walk? The two sides of the rectangle are the legs, and the diagonal is the hypotenuse.

• Square the legs: 8² = 64 and 6² = 36.
• Add them: 64 + 36 = 100, so c² = 100.
• Square root: c = √100 = 10.

  a² + b² = c²
  8² + 6² = c²
  64 + 36 = c²
      100 = c²
        c = √100 = 10

Tips that make the theorem feel easy

• Find the hypotenuse first. It is always the longest side and always sits across from the right angle. Once you know which side is c, the rest falls into place.
• Add for the hypotenuse, subtract for a leg. If c is missing, add the squares. If a leg is missing, the hypotenuse is the biggest square and you subtract from it.
• Don't forget the square root. After adding or subtracting you have a squared value. The √ button turns it back into a real length.
• Remember it only works on right triangles. No 90° corner means a² + b² = c² does not apply.
• Memorize it. The Pythagorean theorem is not on the CAEC formula sheet, so commit "a² + b² = c²" to memory before test day.

Your turn: practice problems

Decide whether you need the hypotenuse or a leg, then solve. Round to one decimal place where needed. No peeking until you have tried.

1. Legs of 6 and 8, find the hypotenuse.
2. A right triangle has a hypotenuse of 10 and one leg of 6, find the other leg.
3. A TV screen is 12 in wide and 5 in tall, find the diagonal.
4. Legs of 7 and 7, find the hypotenuse (round to 1 decimal).