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Pythagorean Theorem Calculator

Calculate missing sides of right triangles using the Pythagorean theorem. Free, fast, accurate — no signup, mobile-frien

Triangle Calculation

Calculation Type

Triangle Sides

Length of side a (leg)

Length of side b (leg)

Length of hypotenuse (longest side)

Options

Number of decimal places

Calculation Results

Side a

3.00

Leg

Side b

4.00

Leg

Hypotenuse c

5.00

Longest side

Right Triangle?

Yes ✓

a² + b² = c²

Triangle Angles

Angle A
36.87°
opposite side a
Angle B
53.13°
opposite side b
Angle C
90.00°
right angle

Triangle Measurements

Area:6.00 square units
Perimeter:12.00 units
Altitude to hypotenuse:2.40 units
Inradius:1.00 units
Circumradius:2.50 units

📐 About the Pythagorean Theorem

Formula: a² + b² = c²

Named after: Pythagoras (c. 570-495 BC), though known earlier

Applies to: Right triangles only (triangles with a 90° angle)

Key insight: The square of the hypotenuse equals the sum of squares of the other two sides

How it works

The Pythagorean theorem links the three sides of a right triangle: the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two. Rearranging it lets you find any one side when you know the other two.

Pythagorean theorem

a² + b² = c²        →  c = √(a² + b²)        and  a = √(c² − b²)
a, b
the two legs (sides forming the right angle)
c
the hypotenuse (longest side)

Worked example

  • Legs a = 3, b = 4
  1. c² = 3² + 4² = 9 + 16 = 25
  2. c = √25

c = 5 — the classic 3-4-5 right triangle.

Good to know

  • It only works for right triangles (one 90° angle) — for other triangles you need the law of cosines.
  • To find a leg instead of the hypotenuse, subtract: a = √(c² − b²).
  • “Pythagorean triples” like 3-4-5, 5-12-13, and 8-15-17 are whole-number right triangles worth recognizing.

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Frequently Asked Questions

What is the Pythagorean theorem?

In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The hypotenuse c is always the longest side, opposite the right angle.

How do I find a missing leg instead of the hypotenuse?

Rearrange the formula: a = √(c² - b²). For example, with a hypotenuse of 13 and one leg of 5, the other leg is √(169 - 25) = √144 = 12.

Does the Pythagorean theorem work for all triangles?

No — only right triangles. For other triangles, use the law of cosines, which generalizes the theorem: c² = a² + b² - 2ab·cos(C).

What are Pythagorean triples?

Whole-number side sets that satisfy the theorem, like (3, 4, 5), (5, 12, 13), and (8, 15, 17). Any multiple of a triple, such as (6, 8, 10), also works.

Where is the Pythagorean theorem used in real life?

Builders square foundations with the 3-4-5 method, and it gives the diagonal of a TV or room, ladder placement distances, and straight-line distance between two points on a coordinate grid.