Pythagorean Theorem Calculator: Find the Missing Side

In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs:

• Solve for the hypotenuse: c = √(a² + b²)
• Solve for leg a: a = √(c² − b²)
• Solve for leg b: b = √(c² − a²)

The theorem only holds for right triangles — one angle must be exactly 90°. But it works the other way too: if a² + b² = c² for a triangle's three sides, it must be a right triangle. That converse is how carpenters square a corner with a 3-4-5 tape check. See the Pythagorean theorem article for the history and the many known proofs.

A ramp is 3 m high and 4 m long at the ground. How long is the sloped surface?

Hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25 = 5 m. Area: 3 × 4 ÷ 2 = 6 m². Perimeter: 3 + 4 + 5 = 12 m. Angles: α = arctan(3/4) ≈ 36.87°, β = 90° − 36.87° ≈ 53.13°.

Now reverse it: a 13 m ladder leans against a wall with its foot 5 m out. Leg a = √(13² − 5²) = √(169 − 25) = √144 = 12 m up the wall.

Identify which side you are missing before you type anything — that is the most common mistake.

Step by step

1. Choose what to solve for. Hypotenuse (c) when you know both legs, or leg a / leg b when you know the hypotenuse and one leg.
2. Enter the two known sides. Any positive length works — use the same unit for both.
3. Press Calculate. You get the missing side, the area, the perimeter, both acute angles and the altitude to the hypotenuse.
4. Check the working. The calculator shows every substitution so you can verify the math.

A Pythagorean triple is a set of three whole numbers that satisfies the theorem — no square roots needed:

Multiples of a triple work too: 6-8-10 is just 2 × (3-4-5). Builders use the 3-4-5 check to square up a foundation — mark 3 ft along one wall and 4 ft along the other, and the diagonal between the marks must be exactly 5 ft.

The missing side is only half the story — a right triangle's other measures follow from the same three numbers:

• Area = a × b ÷ 2 — half the rectangle the two legs form.
• Perimeter = a + b + c — the distance around the triangle.
• Angle α = arctan(a ÷ b) — the angle opposite leg a, in degrees.
• Angle β = 90° − α — the two acute angles always sum to 90°.
• Altitude h = (a × b) ÷ c — the perpendicular drop from the right angle to the hypotenuse.

That altitude result is useful whenever you need a height measured at a right angle to a sloped line, for example cutting a support that meets a sloped beam. If the triangle is part of a ramp or a hill, a slope calculator turns the rise and run into a grade and an angle.

The Pythagorean theorem is one of the most used formulas outside of school:

• Ladders: how far up the wall a ladder of known length reaches from a given footing distance.
• Construction: squaring corners with the 3-4-5 check, and cutting diagonal braces to exact length.
• Screens and rooms: a 16:9 TV's diagonal from its width and height, or the longest item that fits across a room diagonally.
• Navigation: the straight-line distance across a grid when you know the north-south and east-west legs.
• Surveying: turning two perpendicular measurements into a direct distance.

When the triangle you are measuring represents a plot of ground, an area converter converts the resulting area between square feet, square meters, acres and hectares.