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Pythagorean Theorem Calculator

Finds the missing side of a right triangle from any two of the legs a, b and hypotenuse c using a² + b² = c², and reports the area, perimeter and acute angles.

When to use

You know two sides of a right triangle and need the third, or want to check whether three lengths form a right triangle.

Do not use when: The triangle is not right-angled (use triangle-solver) or you need the distance between coordinates (use distance-2d).

Formula

c = √(a² + b²); a = √(c² − b²); b = √(c² − a²); area = a·b/2; angle_a = atan(a/b), angle_b = atan(b/a)

Inputs

Parameter	Type	Required	Description
`a`	number	no	First leg (a side adjacent to the right angle). Give any two of a, b, c. Range: > 0
`b`	number	no	Second leg. Range: > 0
`c`	number	no	Hypotenuse (the longest side, opposite the right angle). Range: > 0

Outputs

Output	Type	Unit	Description
`side_a`	number		Length of leg a.
`side_b`	number		Length of leg b.
`hypotenuse_c`	number		Length of the hypotenuse.
`solved_for`	string		Which side was computed ('a', 'b' or 'c'), or a note that all three were given and consistent.
`area`	number		a × b / 2.
`perimeter`	number		a + b + c.
`angle_a_degrees`	number	°	atan(a / b) in degrees.
`angle_b_degrees`	number	°	atan(b / a) in degrees; the two acute angles sum to 90°.

Example

a = 3, b = 4: {"a":3,"b":4} → {"side_a":3,"side_b":4,"hypotenuse_c":5,"solved_for":"c","area":6,"perimeter":12,"angle_a_degrees":36.8699,"angle_b_degrees":53.1301}

a = 5, c = 13: {"a":5,"c":13} → {"side_b":12,"solved_for":"b","area":30,"perimeter":30,"angle_a_degrees":22.6199}

GET https://tttkmbb.com/api/v1/calculate/pythagorean?a=3&b=4

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/pythagorean (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/pythagorean · Markdown: https://tttkmbb.com/math/pythagorean.md · JSON definition: https://tttkmbb.com/math/pythagorean.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="pythagorean"
OpenAPI operationId: solve_pythagorean_theorem
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Pythagorean theorem (reference)
Wolfram MathWorld – Pythagorean Theorem (reference)

FAQ

What if I enter all three sides?

They are checked against a² + b² = c². Consistent values are returned unchanged; otherwise an error reports the hypotenuse the two legs imply.

Why must the hypotenuse be the longest side?

Since c² = a² + b², c is always longer than either leg. A leg equal to or longer than c gives a negative square and no real triangle.

Related calculators

Right Triangle Calculator — Solve a right triangle from sides and angles.
Distance Between Two Points Calculator — Distance between points uses the same theorem.
Triangle Solver — General triangles (law of sines and cosines).