Pythagorean Theorem Calculator

Solve for any missing side of a right triangle using a² + b² = c². View instant angle calculations, triangle area, perimeter, and Pythagorean triple detection.

Solve For Missing Side

Quick Triples
Applied Formulac = √(a² + b²)
Calculated Hypotenuse c
5units

c = √(3² + 4²)

Pythagorean Triple

Side a (Leg)

3

Side b (Leg)

4

Side c (Hypotenuse)

5

Calculated

Area

6

sq units

Perimeter

12

units

Angle A (α)

36.87°

opposite leg a

Angle B (β)

53.13°

opposite leg b
Right Triangle Geometric Profileb = 4a = 3c = 5
Fundamental Pythagorean Theorem Formula
Standard Equationa² + b² = c²
Find Hypotenuse (c)c = √(a² + b²)
Find Leg (a or b)a = √(c² − b²)
Step-by-Step Calculation Breakdown
Example 1: Finding Hypotenuse with a = 6, b = 8
c=√(6² + 8²)=√(36 + 64)=√100=10.00 (Pythagorean Triple 3-4-5 × 2)
Example 2: Finding Leg a with b = 12, c = 13
a=√(13² − 12²)=√(169 − 144)=√25=5.00

Real-World Applications

  • Carpentry & Construction: Square up walls, foundations, and decks with the 3-4-5 rule.
  • Roofing & Framing: Calculate rafter lengths and roof pitch diagonals.
  • Navigation & Mapping: Calculate straight-line Euclidean distance between Cartesian coordinate points.

Frequently Asked Questions

What is the Pythagorean theorem formula?
The Pythagorean theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse (c) is equal to the sum of the areas of the squares on the other two sides (a and b): a² + b² = c².
How do you calculate the hypotenuse (c)?
To find hypotenuse c given legs a and b, use c = √(a² + b²). For example, with legs 3 and 4: c = √(9 + 16) = √25 = 5.
How do you find a missing leg (a or b)?
To solve for a missing leg, rearrange the equation: a = √(c² − b²) or b = √(c² − a²). The hypotenuse (c) must always be strictly longer than either leg.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that perfectly satisfy a² + b² = c². Examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

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