Pythagoras Triangle - Cathetus and Hypotenuse Solver

A pythagoras triangle is any right triangle, the kind with one 90 degree corner. The two perpendicular sides are called the catheti, c1 and c2, and the longest side opposite the right angle is the hypotenuse, h. This calculator lets you enter any two of those three sides and returns the third side, the area, and the perimeter, all using the Pythagorean theorem c1^2 + c2^2 = h^2.

• • Roof rafters and stair stringers: Find the hypotenuse and perimeter of a rafter or stair stringer from the run and rise.
• • Ladder and reach problems: Find how high a ladder of known length reaches a wall, from the length of the hypotenuse and one cathetus.
• • Slope and grade checks: Convert a measured slope distance and a known rise into a horizontal leg, for drainage, ramps, and grading.
• • Geometry homework and exam problems: Solve pythagoras triangle problems written in the European cathetus and hypotenuse notation without translating the variables first.

The right triangle is named after the Greek mathematician Pythagoras, whose theorem ties the three sides together. The 90 degree angle forces the two catheti to be perpendicular, which is why the area simplifies to half the product of the catheti and why the hypotenuse is always the longest side.

The cathetus and hypotenuse naming is common in European geometry, surveying, and carpentry, and it shows up in many textbooks. This calculator keeps that naming visible while still using the same a^2 + b^2 = c^2 rule that links all right triangles.

For a right triangle page that uses the more familiar side a, side b, and side c labels, the Right Triangle Calculator is the closest peer and walks through the same five-value solve.

The calculator reads the catheti c1, c2, and hypotenuse h fields, decides which two sides the user knows, and applies the matching form of the Pythagorean theorem. With all three sides known, it computes the area and perimeter in a single pass.

• c1: first cathetus of the right triangle, the side next to the 90 degree angle, measured in the same length unit as the other sides
• c2: second cathetus of the right triangle, the other side next to the 90 degree angle, in the same length unit as c1 and h
• h: hypotenuse, the longest side opposite the 90 degree angle, in the same length unit as the catheti

When both catheti are known, the hypotenuse comes straight from the square root of c1^2 + c2^2. When the hypotenuse and one cathetus are known, the other cathetus comes from the same rule rearranged, c_i = sqrt(h^2 - c_j^2). The area uses (c1 * c2) / 2 because the catheti are perpendicular, and the perimeter is the sum of the three sides.

According to Wolfram MathWorld, the Pythagorean theorem states that for a right triangle with catheti c1 and c2 and hypotenuse h, c1^2 + c2^2 = h^2

To check whether three integer catheti and hypotenuse values actually satisfy c1 squared plus c2 squared equals h squared before applying the theorem, the Pythagorean Triples Calculator verifies the relationship first.

These four ideas cover the rules the calculator uses to recover any missing side and the area and perimeter of a right triangle.

These four facts are the entire toolbox this calculator needs. They are also why the right triangle is so useful in real work: with one known side, every other side and the area can be recovered from just two of c1, c2, h.

When the same problem also needs the area of a triangle that is not a right triangle, the Triangle Area Calculator covers the general base-times-height and Heron cases in one place.

Pick any two of the three sides of the right triangle you already know, type them in, and the result panel fills in the rest of the triangle.

1. 1 Pick a single length unit: Use meters, feet, inches, or any one unit for every side input. Mixing units in a right triangle is the most common cause of a wrong hypotenuse.
2. 2 Enter the two catheti when you have them: Type the run into Cathetus c1 and the rise into Cathetus c2, and leave Hypotenuse h blank. The calculator returns h, the area, and the perimeter.
3. 3 Enter a cathetus and the hypotenuse when those are known: Type the long side into Hypotenuse h and the perpendicular leg into Cathetus c1 (or c2). The missing cathetus, area, and perimeter appear together.
4. 4 Read the solved triangle: Use the hypotenuse for the rafter or ladder length, the area for material coverage, and the perimeter for fence, trim, or border length.
5. 5 Reset when the next problem starts: Press Reset to clear the inputs and return to the default 3-4-5 right triangle. The result panel updates in real time on every change.

Suppose a stair stringer must rise 2.4 m over a horizontal run of 1.0 m. The run is cathetus c1, the rise is cathetus c2, and the stringer is the hypotenuse. Enter c1 = 1.0 and c2 = 2.4. The hypotenuse h is 2.6 m, the area of the right triangle under the stringer is 1.2 square meters, and the perimeter is 6.0 m. That is the same 5-12-13 ratio scaled to the run-and-rise of the stair.

If the side you need to find belongs to a non-right triangle such as a sloped roof run, the Triangle Length Calculator handles side-length problems for general triangle shapes.

The cathetus and hypotenuse naming is the most compact way to write a right triangle, and this calculator keeps the entire solution visible at once.

• • Solves the third side from any two sides: Two catheti, or a cathetus plus the hypotenuse, both return the missing side instantly. You do not have to rearrange the problem to match the tool.
• • Returns area and perimeter alongside the sides: Cathetus c1, cathetus c2, hypotenuse h, area, and perimeter all appear together, so the pythagoras triangle answer can feed a follow-up step without re-entering values.
• • Matches European textbook notation: Cathetus and hypotenuse are the standard labels in European geometry, surveying, and carpentry. The result lines up with the variables you have written down.
• • Decimal-friendly input: Field-measured lengths such as 3.27 m or 4.05 m work without rounding. The Pythagorean calculation is done at full precision and only the final display is rounded.
• • Guards against impossible inputs: The hypotenuse must be longer than either cathetus for a valid right triangle. Inputs that violate the rule are rejected with a clear error message.

If the right triangle is part of a larger shape, the perimeter feeds fence, trim, or border work, while the area feeds material coverage such as flooring, fabric, or sheathing. The hypotenuse also doubles as the diagonal of a rectangle with the same catheti, useful for checking square layouts on site.

For a quick area of a rectangle, circle, or any other shape that the catheti and hypotenuse feed into, the Area Calculator keeps the rest of the area math in one place.

A few input choices decide whether the solved triangle matches the actual shape you are working with.

• • Only two known sides are needed to solve a right triangle, but the inputs must include at least one cathetus or the hypotenuse. The result panel will ask for two of c1, c2, h if only one is provided.
• • The Pythagorean theorem is for right triangles only. If the actual shape is scalene or obtuse, use the general triangle calculator instead, which accepts base-height, SAS, or SSS inputs.
• • Rounded output can differ from a hand calculation by a few hundredths of a unit. Keep full precision through the calculation, then round only the final displayed numbers for a worksheet answer.

If your sides come from a drawing, double-check whether the values are edge-to-edge or include a wall, fence, or material thickness. Stripping that thickness before entering the sides keeps the solved right triangle aligned with the shape being measured.

According to Math Open Reference, a right triangle has one 90 degree angle, the two perpendicular sides are the catheti, and the side opposite the right angle is the hypotenuse

According to Wikipedia, the area of a right triangle equals half the product of the two catheti because the catheti are perpendicular