In a right triangle, the squares on the two shorter sides add to the square on the longest.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Written down
a² + b² = c², where c is the hypotenuse, the side opposite the right angle.
Right triangles only
This is not true of any other triangle. Check for the right angle before using it.
Finding a shorter side
If c and one leg are known, subtract instead: a² = c² − b².
The statement, precisely
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c². The hypotenuse is the side opposite the right angle and is always the longest.
It is a statement about areas
Draw an actual square on each side. The two smaller squares, cut up, fit exactly inside the largest. The theorem is a fact about area, which is why the sides appear squared, and seeing that once makes the formula memorable.
It only holds for right angles
The relationship fails for any other triangle. Conversely, if a² + b² = c² holds, the triangle must have a right angle — which is how builders check a corner is square using a 3-4-5 triangle.
Finding a shorter side
If c is known and a is missing, rearrange: a² = c² − b². Subtracting rather than adding is the case people get wrong, so identify the hypotenuse before writing anything — it is always opposite the right angle.
Step 2: Try It Yourself
Tap and try it out.
Area · squares inside
12
Perimeter · walk around
14
3 × 4 = 12 · 3 + 4 + 3 + 4 = 14
Area counts the units inside. Perimeter measures the walk all the way round.
Step 3: Watch an Example
One step at a time.
Watch Leo Find a Hypotenuse
A right triangle has legs 6 and 8.
- Step 1
He squares both legs: 36 and 64.
Step 4: Your Turn
Practice makes it stick.
The Ladder
Problem 1 of 2
A right triangle has legs 3 and 4. What is the hypotenuse?
The Missing Leg
Problem 2 of 2
A right triangle has hypotenuse 13 and one leg 5. What is the other leg?
Find the Side
1 of 4
Legs 5 and 12. Hypotenuse?
2 of 4
Legs 8 and 15. Hypotenuse?
3 of 4
Hypotenuse 10, one leg 6. Other leg?
4 of 4
Does the theorem work for any triangle?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Legs 9 and 12. Hypotenuse?
Question 2 of 2
Which side is c in a² + b² = c²?
What You Learned
- In a right triangle, a² + b² = c².
- c is the hypotenuse, opposite the right angle.
- It works only for right triangles.