Pythagorean Theorem Calculator
Enter any two sides of a right triangle and the calculator finds the third using a² + b² = c². It also gives the area, perimeter, the two acute angles and a scaled diagram.
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How to use Pythagorean Theorem Calculator
- Fill in any two of side a, side b and the hypotenuse c.
- Leave the third box empty; it is the one that will be solved.
- Read all three sides, the area, perimeter and angles.
- Check the diagram and the written steps.
The theorem in practice
The theorem links the three sides of any right triangle. If the legs are a and b, then c = √(a² + b²). If you know the hypotenuse and one leg, the other leg is √(c² − leg²).
Worked example: legs 3 and 4 give c = √(9 + 16) = √25 = 5, the classic 3-4-5 triangle. Its area is ½ × 3 × 4 = 6, its perimeter is 12, and angle A = arctan(3 / 4) ≈ 36.87°, so angle B ≈ 53.13°. Another whole-number example is 5-12-13.
Typical uses
- Checking that a corner is square in construction using the 3-4-5 method.
- Finding the length of a ladder, ramp or brace.
- Calculating the diagonal of a rectangle or screen.
- Distance between two points on a grid.
- Homework and exam practice on right triangles.
Notes and limits
The theorem works only for right triangles. If the triangle has no right angle, use the law of cosines instead. The tool needs exactly two known sides; if you know an angle and one side you will need trigonometry. Results are shown to about six decimal places. Because the diagram is drawn in proportion, very thin triangles are given a minimum size so labels stay readable. All calculations run in your browser. To check the result by eye, the hypotenuse must always be longer than either leg but shorter than the sum of the two legs. Surveyors and carpenters also use multiples of the 3-4-5 triangle, such as 6-8-10 or 9-12-15, because every multiple of a right triangle is still a right triangle. The altitude to the hypotenuse, which the tool also reports, equals the product of the legs divided by the hypotenuse.
Why use this pythagorean theorem calculator tool
- Solve any side. Find the hypotenuse from the legs, or a leg from the hypotenuse and the other leg.
- Angles and area. Also shows both acute angles in degrees, the area, the perimeter and the altitude to the hypotenuse.
- Scaled diagram. An SVG triangle drawn in proportion to your values with every side labelled.
- Validated input. Rejects zero, negative or impossible values, such as a hypotenuse shorter than a leg.
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². The hypotenuse is the side opposite the right angle and is always the longest.
How are the angles found?
Angle A, opposite side a, equals arctan(a / b). Angle B is 90° minus A, because the angles of a triangle sum to 180°.
Does the unit matter?
Use the same unit for all sides. The area is in that unit squared, and the angles are not affected by the unit.
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