Pythagorean Theorem Calculator
In a right triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. This calculator finds the missing side, then gives the area, the perimeter and the two acute angles.
How to use the pythagorean theorem calculator
- Choose whether you want the hypotenuse or a missing leg.
- For the hypotenuse, enter the two legs. For a leg, enter the hypotenuse in the first box and the known leg in the second.
- Read the missing side, area, perimeter and angles. Units are whatever you entered, as long as they match.
Formula
a² + b² = c², so c = √(a² + b²) and b = √(c² − a²). The acute angles come from trigonometry: angle A = arctan(a ÷ b) and angle B = 90° − A. Area = ½ × a × b.
Worked examples
The 3-4-5 triangle
Legs 3 and 4 give c = √(9 + 16) = √25 = 5. The area is ½ × 3 × 4 = 6 and the perimeter is 12.
Finding a leg
A 13 ft ladder leans against a wall with its foot 5 ft from the wall. The height it reaches is √(13² − 5²) = √144 = 12 ft.
Scaling
Doubling the 3-4-5 sides gives 6-8-10, so c = 10. Any multiple of a Pythagorean triple is another triple.
Using the Pythagorean theorem
Pythagoras' theorem connects the three sides of every right triangle, and it is one of the most used results in geometry. Builders use it to check that corners are square: measure 3 units along one wall and 4 along the other, and the diagonal must be exactly 5.
Common Pythagorean triples
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 20 | 21 | 29 |
Distance between two points
The distance formula is the theorem in disguise. For points (x₁, y₁) and (x₂, y₂), the horizontal and vertical differences are the legs of a right triangle and the distance is the hypotenuse: d = √((x₂ − x₁)² + (y₂ − y₁)²).
Practical uses
Finding the length of a ladder needed to reach a window, the diagonal of a TV screen from its width and height, the shortest path across a rectangular field, or the straight-line distance on a map grid.
Special right triangles
The 45-45-90 triangle has sides in the ratio 1 : 1 : √2, and the 30-60-90 triangle has sides in the ratio 1 : √3 : 2. If you recognise them you can skip the calculation.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
Hypotenuse c: 15
- A 17 ft ladder reaches 15 ft up a wall. How far is its base from the wall?
Show answer
Missing leg b: 8
Common mistakes to avoid
- Using the theorem on a triangle that has no right angle.
- Adding the sides instead of their squares.
- Entering a hypotenuse shorter than a leg. The hypotenuse is always the longest side.
Frequently asked questions
What is the Pythagorean theorem?
For a right triangle with legs a and b and hypotenuse c, a² + b² = c².
What are Pythagorean triples?
Sets of whole numbers that satisfy the theorem, such as 3-4-5, 5-12-13 and 8-15-17.
Does it work in three dimensions?
Yes. The diagonal of a box with sides a, b and c is √(a² + b² + c²).
Can I use different units?
Use the same unit for all sides. The area comes out in square units of that unit.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.