Pythagorean Theorem Calculator: Find Missing Side, Area & Perimeter

How to use the tool

1. Fill two fields
   Example 1: Side A = 5.6, Side B = 7.1.
   Example 2: Side A = 11, Side C = 15.
2. Select units — centimeters for Example 1, feet for Example 2.
3. Set decimal precision — 3 places (Example 1) or 2 places (Example 2).
4. Press Calculate to display the missing side, area and perimeter.

Formulas applied

• Hypotenuse: $$c = \sqrt{a^{2}+b^{2}}$$
• Missing leg: $$a = \sqrt{c^{2}-b^{2}},\; b = \sqrt{c^{2}-a^{2}}$$
• Area: $$A = rac{1}{2}ab$$
• Perimeter: $$P = a + b + c$$

Checked examples

• Example 1 — Inputs 5.6 cm & 7.1 cm → c = 9.042 cm; A = 19.880 cm²; P = 21.742 cm.
• Example 2 — Inputs 11 ft & 15 ft → b = 10.20 ft; A = 56.09 ft²; P = 36.20 ft.

Quick-Facts

• Proof of $$c^{2}=a^{2}+b^{2}$$ recorded in Euclid’s Elements I.47 (Euclid, ~300 BCE).
• Babylonian tablet Plimpton 322 lists Pythagorean triples (~1800 BCE) (Robson, 2001).
• Builders square corners with the 3-4-5 rule (Fine Woodworking, 2018).
• Right-triangle area equals 0.5 ab (University of Colorado Boulder, 2023).

FAQ

What does the calculator output?

It returns the complete side set, right-triangle area and perimeter instantly once you supply any two sides.

Which side is which?

Sides A and B are the legs adjacent to the right angle; Side C is the hypotenuse opposite it (Euclid, Elements I.47).

How do I confirm my inputs form a right triangle?

The script re-checks $$c^{2}=a^{2}+b^{2}$$; if it fails, you see an error message instead of results.

What units can I work in?

You choose meters, centimeters, inches or feet; outputs follow the same unit so no manual conversions.

Why can’t I enter zero or negative lengths?

A triangle’s side length is strictly positive by definition (Weisstein, MathWorld).

How accurate is the calculation?

The browser uses IEEE-754 double precision, giving roughly 15-digit accuracy (IEEE Std 754-2019).

Can I derive a roof slope from the results?

Yes—divide rise (vertical leg) by run (horizontal leg), then convert to pitch or angle as needed (OSHA Roof Guidelines, 2021).

Does the Pythagorean theorem hold in curved space?

No; non-Euclidean geometries replace it with cosine-rule variants (Milnor, Hyperbolic Geometry, 1982).