Pythagorean Theorem Calculator

The Pythagorean theorem says that in a right triangle, a² + b² = c², where c is the hypotenuse. Enter any two sides and the calculator finds the third, showing each step and the exact answer as a simplified square root (such as 5√2) as well as a decimal. It also draws the triangle to scale, works out the angles and area, and can check whether three lengths make a right triangle.

The Pythagorean theorem

In any right triangle — a triangle with one 90° angle — the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, called the legs: a² + b² = c². The theorem is named after the Greek mathematician Pythagoras (about 570–495 BC), although the relationship was known and used centuries earlier in Babylon, Egypt, India and China. It is one of the most useful results in mathematics, with hundreds of known proofs.

Formulas for each side

How to use the calculator

1. Enter the two sides you know and leave the third box empty.
2. Use the same unit for both — cm, m, inches, feet or none.
3. Read the missing side as an exact root and a decimal.
4. Follow the steps to see the working, which you can copy into homework.
5. Check the angles, area, perimeter and the scale drawing.
6. To test whether three lengths form a right triangle, fill in all three boxes.

Worked example: a ladder against a wall

A 5 m ladder leans against a wall with its foot 1.5 m from the wall. How high up the wall does it reach? The ladder is the hypotenuse (5 m) and the distance from the wall is one leg (1.5 m). The height is √(5² − 1.5²) = √(25 − 2.25) = √22.75 ≈ 4.77 m. Enter a = 1.5 and c = 5, leave b blank, and the calculator shows the same steps with the angle between ladder and ground — about 72.5°, a little shallower than the roughly 75° usually recommended for ladders (the 1-in-4 rule: one unit out for every four units up).

Simplifying square roots

When the answer isn’t a whole number, it can be written exactly as a simplified square root. Look for the largest perfect square that divides the number: √50 = √(25 × 2) = 5√2 and √72 = √(36 × 2) = 6√2. So a right triangle with legs 5 and 5 has a hypotenuse of exactly 5√2 ≈ 7.071. Exact forms are expected in many maths courses; decimals are more practical for measuring.

Pythagorean triples

A Pythagorean triple is a set of three whole numbers that fit a² + b² = c², such as 3, 4, 5 or 5, 12, 13. Any multiple of a triple is also a triple — 6, 8, 10 or 9, 12, 15 — but the “primitive” triples have no common factor. Euclid’s formula generates all primitive triples: choose whole numbers m > n that have no common factor and are not both odd, then a = m² − n², b = 2mn and c = m² + n². The list in the calculator shows the 16 primitive triples with a hypotenuse up to 100.

Real-world uses

• Building: the 3-4-5 rule checks that corners are square — measure 3 ft along one wall, 4 ft along the other, and the diagonal should be 5 ft.
• Screens: TV and monitor sizes are diagonals; a 16:9 screen 48 in wide and 27 in tall is about 55 in diagonally.
• Navigation: the straight-line distance for 3 km north and 4 km east is 5 km.
• Ramps and roofs: rise and run give the sloped length.
• Distance between points: the distance formula in coordinate geometry is the Pythagorean theorem.

Pythagoras in three dimensions

The theorem extends to three dimensions. The longest diagonal of a box with sides a, b and c has length √(a² + b² + c²). For a room 4 m long, 3 m wide and 2.4 m high, the diagonal from one floor corner to the opposite ceiling corner is √(16 + 9 + 5.76) = √30.76 ≈ 5.55 m — useful when checking whether a long object will fit. Apply the two-dimensional theorem twice: first across the floor (5 m), then up to the ceiling.

Common mistakes

• Using the hypotenuse as a leg — c is always the longest side, opposite the right angle.
• Adding instead of subtracting when finding a leg.
• Forgetting to take the square root at the end.
• Mixing units, such as feet and inches — convert first.
• Applying the theorem to triangles without a right angle.

The converse and the triangle checker

The converse of the theorem is also true: if a² + b² = c² for the three sides of a triangle, then the triangle has a right angle. If a² + b² is greater than c² (with c the longest side) the triangle is acute; if it is less, the triangle is obtuse. The checker uses this to classify any three lengths — and tells you if they can’t form a triangle at all.