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Geometry

Pythagorean Theorem

Calculate hypotenuse or legs. Free online Pythagorean Theorem. Calculate pythagorean theorem online — fast, accurate, mobile-friendly, no signup needed.

a² + b² = c² where c is the hypotenuse.
Hypotenuse c
5

Derivation

  1. ├── 01Givena = 3, b = 4
  2. ├── 02Formula√(t²+a²)
  3. ├── 03Substitute√(t²+3²)
  4. └── 04Compute Hypotenuse c5
Did you know?

Pythagoras of Samos (c. 570–495 BCE) likely wasn’t the first to find a²+b²=c² — Babylonian tablets from ~1800 BCE used it — but the first rigorous proof is credited to his school.

§01What is

Understanding the Pythagorean Theorem

The Pythagorean Theorem computes Hypotenuse c from 2 inputs: leg a, leg b. Calculate hypotenuse or legs.

Geometry is what turns raw measurements into useful answers about space — how much paint, how big a yard, how much material a project will need. Every craftsperson, architect, and DIYer reaches for these formulas regularly. The Pythagorean Theorem sits in that toolkit — it calculate hypotenuse or legs. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

√(t²+a²)

Where

a
Leg a
b
Leg b
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: Leg a = 3, Leg b = 4.

  1. 01Start by noting the input — Leg a: 3.
  2. 02Start by noting the input — Leg b: 4.
  3. 03Substitute these values into the formula: √(t²+a²)
  4. 04Compute Hypotenuse c: the calculator returns 5.
  5. 05Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Pythagorean Theorem Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

Leg a halved

a = 1.5 (from 3)

Keep every other input at its default and halve the leg a. See how hypotenuse c responds.

  1. 01New Leg a: 1.5
  2. 02Baseline Hypotenuse c: 5
  3. 03New Hypotenuse c: 4.272
  4. 04Hypotenuse c decreases by 14.6% → use this sensitivity to plan for real-world variation.
02 · PATTERN

Leg a doubled

a = 6 (from 3)

Keep every other input at its default and double the leg a. See how hypotenuse c responds.

  1. 01New Leg a: 6
  2. 02Baseline Hypotenuse c: 5
  3. 03New Hypotenuse c: 7.2111
  4. 04Hypotenuse c increases by 44.2% → use this sensitivity to plan for real-world variation.
03 · PATTERN

Leg b halved

b = 2 (from 4)

Keep every other input at its default and halve the leg b. See how hypotenuse c responds.

  1. 01New Leg b: 2
  2. 02Baseline Hypotenuse c: 5
  3. 03New Hypotenuse c: 3.60555
  4. 04Hypotenuse c decreases by 27.9% → use this sensitivity to plan for real-world variation.
04 · PATTERN

Leg b doubled

b = 8 (from 4)

Keep every other input at its default and double the leg b. See how hypotenuse c responds.

  1. 01New Leg b: 8
  2. 02Baseline Hypotenuse c: 5
  3. 03New Hypotenuse c: 8.544
  4. 04Hypotenuse c increases by 70.9% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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