Pythagorean Theorem Calculator

Right-triangle sides, area, and acute angles

The main value is the requested missing side, or the calculated right-triangle hypotenuse in checking mode. Area, perimeter and acute angles describe the right-triangle reference formed by the two entered legs. The comparison reports how the measured longest side differs from that reference.

To put both sides into matching units first, use the Length Converter before applying the right-triangle formula.

The formula

The legs a and b meet at the right angle, while c is the hypotenuse opposite it. Triangle area is ab/2 and perimeter is a + b + c. The angle opposite the entered known leg is arctan(a/b), expressed in degrees.

How to use this calculator

1. Choose hypotenuse, missing leg or a three-side check, and the common length unit.
2. Enter the labeled sides. Solving modes carry the same valid reference triangle when you switch tasks.
3. For checking, enter the measured longest side and a relative-length tolerance. Read the live calculated hypotenuse and comparison.

Identify the hypotenuse first

The hypotenuse is always opposite the right angle and is the longest side. In missing-leg mode, the second input must therefore exceed the known leg. Equal values would produce a zero-length leg, not a proper triangle.

In three-side checking mode the longest side must form a nondegenerate triangle with the other two. Its relative-length difference from √(a² + b²) is compared with your chosen tolerance. A close match supports approximate right-triangle consistency; it does not establish measurement certainty.

Use the result for a diagonal or a height

A rectangle’s diagonal forms a right triangle with its width and height. Enter those two perpendicular dimensions as legs to obtain the diagonal. The same idea can apply to a vertical rise and horizontal run.

The length selector converts visible measurements, including squared area scaling in the results. In checking mode, the area and angles remain those of the perpendicular-leg reference, rather than a general triangle solved from the measured three sides. Use the triangle calculator for actual non-right triangle geometry.

A tolerance for measured sides

With legs 3 m and 4 m, the reference hypotenuse is 5 m. A measured longest side of 5.002 m differs by 0.04%. It is within a 0.1% tolerance and outside a 0.03% tolerance. This comparison uses length difference, not a squared-side residual or angular tolerance.

Checking keeps the measured longest side when you leave and return to that mode. The first visit starts from the current reference hypotenuse if a valid solving result is available. Reset restores legs 3 and 4, meters and the 0.1% comparison tolerance.

Assumptions & limitations

What this calculation assumes

• The triangle is flat and has one angle of exactly 90°.
• Both entered sides are positive and share one length unit.

What to keep in mind

• Non-right triangles require additional information and a different formula.
• The calculator does not account for measurement tolerances or curved surfaces.
• Tolerance checking compares lengths, not uncertainty or an angle. Other results in checking mode describe the right-triangle reference.

Common questions