The Pythagorean Theorem lets you find a missing side of any right triangle. It shows up constantly on the GED — in ladders, diagonals, ramps, screens, and distances.
What Is the Pythagorean Theorem?
The Pythagorean Theorem works on right triangles only — triangles that have a 90° angle. It says that the squares of the two shorter sides add up to the square of the longest side.
a² + b² = c²
The three sides:
• a and b are the two legs — the shorter sides that form the right angle
• c is the hypotenuse — always the longest side, always across from the right angle
• The right angle is marked with a small square in the corner
Finding the Hypotenuse (c)
When you know both legs, you can find c by adding their squares and taking the square root.
1
Write the formula — a² + b² = c²
2
Plug in both legs — replace a and b with the numbers
3
Square each leg — multiply each number by itself
4
Add the squares — this gives you c²
5
Take the square root — this gives you c
Worked Example — Find the Hypotenuse
A right triangle has legs of 6 ft and 8 ft. Find c.
→ Write the formula: a² + b² = c²
→ Plug in: 6² + 8² = c²
→ Square each: 36 + 64 = c²
→ Add: 100 = c²
→ Take the square root: c = √100
c = 10 ft
Calculator shortcut: √(6² + 8²)
Common mistake: The answer is NOT a² + b². You must take the square root at the end. √100 = 10, not 100.
Finding a Missing Leg (a or b)
When you know the hypotenuse and one leg, you can find the other leg by subtracting — then taking the square root.
1
Write the formula — a² + b² = c²
2
Plug in the hypotenuse and the known leg
3
Square both numbers
4
Subtract the known leg² from c²
5
Take the square root — this is the missing leg
Worked Example — Find a Missing Leg
A right triangle has a hypotenuse of 15 ft and one leg of 9 ft. Find the missing leg.
→ Write the formula: a² + b² = c²
→ Plug in: a² + 9² = 15²
→ Square: a² + 81 = 225
→ Subtract 81 from both sides: a² = 225 − 81 = 144
→ Take the square root: a = √144
a = 12 ft
Calculator shortcut: √(15² − 9²)
🎯 Which formula do you use?
→Finding c (hypotenuse): ADD the squares — √(a² + b²)
→Finding a leg: SUBTRACT the squares — √(c² − known leg²)
→Always: The square root is the final step — never skip it
→Check: The hypotenuse must always be the longest side