Pythagoras Triangle Calculator
Calculate the hypotenuse, angles, area, perimeter, altitude, circumradius, and inradius based on two perpendicular side lengths. Pythagoras Triangle Calculator helps students, engineers, builders, and educators solve right-triangle measurements accurately.
Highlights
- Calculate the hypotenuse from two perpendicular side lengths.
- Find both acute angles of a right triangle.
- Compute area and perimeter with ease.
- Determine altitude, circumradius, and inradius.
- Supports multiple length units, including metric and imperial.
Table of Contents
How Pythagoras Triangle Calculator Works?
The Pythagoras Triangle Calculator assumes that Side A and Side B are the two perpendicular legs of a right triangle. From these two positive side lengths, it determines the hypotenuse, the two acute angles, area, perimeter, altitude to the hypotenuse, circumradius, and inradius.
Here, a represents Side A, b represents Side B, and c is the hypotenuse, which is the side opposite the right angle. The acute angles are calculated with inverse tangent relationships:
The area and perimeter use the standard right-triangle relationships:
The altitude from the right angle to the hypotenuse is found by equating the triangle's area in two ways, giving h = ab/c. For a right triangle, the circumcenter is at the midpoint of the hypotenuse, so the circumradius is c/2. The inradius is calculated from the semiperimeter relationship:
| Output | Formula |
|---|---|
| Hypotenuse | c = √(a² + b²) |
| Angle A | tan−1(a/b) |
| Angle B | tan−1(b/a) |
| Area | ab/2 |
| Perimeter | a + b + c |
| Altitude to Hypotenuse | ab/c |
| Circumradius | c/2 |
| Inradius | (a + b − c)/2 |
For example, with Side A equal to 3 and Side B equal to 4, the hypotenuse is 5, the angles are approximately 36.87° and 53.13°, the area is 6 square units, the perimeter is 12 units, the altitude is 2.4 units, the circumradius is 2.5 units, and the inradius is 1 unit.
Practical Use Cases
Students: Students can enter two known perpendicular sides to check homework, study the Pythagorean theorem, and compare calculated angles, area, and circle radii with their own solutions.
Engineers: Engineers can use the calculator for quick right-triangle checks when reviewing diagonal lengths, structural layouts, slopes, and geometric relationships during preliminary design work.
Builders: Builders can calculate a diagonal or verify a right angle from two measured sides, helping with framing, roof layouts, stair planning, and other construction tasks involving perpendicular dimensions.
Educators: Educators can demonstrate how one pair of perpendicular side lengths determines several related measurements, including the hypotenuse, acute angles, area, perimeter, altitude, circumradius, and inradius.
Accuracy and Limitations
The calculator applies standard right-triangle formulas using floating-point arithmetic and accepts positive Side A and Side B values from 0.000001 to 1,000,000,000. Results are mathematically accurate for triangles in which the two entered sides are truly perpendicular, but measured or rounded input values can affect the outputs. It does not solve general non-right triangles, and zero, negative, missing, or out-of-range values are not valid inputs. Extremely large or extremely small values may also be subject to normal computer floating-point precision limits.
Quick Examples
Example #1: Classic 3-4-5 Triangle
Use familiar perpendicular side lengths to calculate the measurements of a standard right triangle.
Example #2: Construction Triangle
Calculate right-triangle measurements for a construction layout using larger side lengths in meters.
Example #3: Small Precision Triangle
Determine measurements for a compact right triangle using millimeter dimensions for precision work.
