Pythagorean Theorem Calculator

Solve for the hypotenuse or legs of a right-angled triangle using the theorem $a^2 + b^2 = c^2$.

Math Audited
Dynamic Proportions Draft
Exact Radical Surd Output
Triangle Inequality Checks
3
1100
4
1100
5
Solved Side Length (c)
5
Triangle Proportions Draft
a: 3b: 4c: 5α: 36.87°β: 53.13°
Area
6
Perimeter
12
Altitude
2.4
Angle γ
90.00°
1. Pythagorean formula:
c = √(a² + b²)
2. Substitute values:
c = √(3² + 4²)
3. Evaluate squares:
c = √(9 + 16)
4. Sum elements:
c = √(25)
6. Final decimal representation:
c ≈ 5
Scientific References & Assumptions
Assumptions:
  • The triangle is planar (2D Euclidean space).
  • The angle between legs a and b is exactly 90 degrees.
  • Side lengths are represented in the same linear units.
Sources & Citations:

Frequently Asked Questions

What is the formula for the Pythagorean theorem?

The Pythagorean theorem formula is a² + b² = c², where a and b are the lengths of the two shorter sides (legs) of a right-angled triangle, and c is the length of the longest side (the hypotenuse).

How do you find the hypotenuse of a right triangle?

To find the hypotenuse c, square the lengths of the legs a and b, add the squares together, and take the square root of the sum: c = √(a² + b²).

Can you use the Pythagorean theorem on any triangle?

No, the Pythagorean theorem only works on right-angled triangles (triangles where one angle is exactly 90 degrees). For oblique triangles, you must use the Law of Sines or the Law of Cosines.

What are the common Pythagorean triples?

Pythagorean triples are sets of three positive integers (a, b, c) that perfectly satisfy the formula a² + b² = c². The most common primitive triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).

How do you check if three side lengths form a right triangle?

To check if three sides form a right triangle, square all three side lengths. Add the squares of the two smaller sides. If their sum equals the square of the largest side, the triangle contains a right angle.

What is a real-life example of the Pythagorean theorem?

A common real-life example is leaning a ladder against a wall. The ladder is the hypotenuse, the ground distance to the wall is leg a, and the height on the wall is leg b. Squaring ground distance and wall height gives the square of the ladder's length.

What is the difference between a primitive and non-primitive triple?

A primitive Pythagorean triple consists of three positive integers that share no common factors other than 1, like (3, 4, 5). A non-primitive triple is a multiple of a primitive one, such as (6, 8, 10), where all sides are multiplied by 2.

Who actually invented the Pythagorean theorem?

While named after the ancient Greek philosopher Pythagoras, the relationship between right triangle sides was known much earlier. Ancient Babylonian, Egyptian, Chinese, and Indian mathematicians documented these triples centuries before Pythagoras lived.

About the Pythagorean Theorem Calculator

The Pythagorean theorem is geometry's most famous rule: in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. This calculator solves for a, b, or c instantly, showing step-by-step equations and interactive triangle visualizations.

Mathematical Formula & Logic

For a right triangle with legs a and b and hypotenuse c, the relation holds: - c = √(a² + b²) (Solve for Hypotenuse) - a = √(c² - b²) (Solve for Leg a) - b = √(c² - a²) (Solve for Leg b) Additional derived metrics: - Area = 0.5 × a × b - Perimeter = a + b + c - Altitude to Hypotenuse = (a × b) / c

Step-by-Step Example

Given legs a = 3 and b = 4, the hypotenuse c is calculated as: - c = √(3² + 4²) = √(9 + 16) = √25 = 5 - Area: 0.5 × 3 × 4 = 6 - Perimeter: 3 + 4 + 5 = 12 - Altitude: (3 × 4) / 5 = 2.4 Given leg a = 6 and hypotenuse c = 10, leg b is calculated as: - b = √(10² - 6²) = √(100 - 36) = √64 = 8 - Area: 0.5 × 6 × 8 = 24 - Perimeter: 6 + 8 + 10 = 24 - Altitude: (6 × 8) / 10 = 4.8

Reference Data & Values

tripletypemultipliersides
(3, 4, 5)Primitive1a = 3, b = 4, c = 5
(6, 8, 10)Non-Primitive2a = 6, b = 8, c = 10
(5, 12, 13)Primitive1a = 5, b = 12, c = 13
(8, 15, 17)Primitive1a = 8, b = 15, c = 17
(7, 24, 25)Primitive1a = 7, b = 24, c = 25

Frequently Asked Questions

The Pythagorean theorem formula is a² + b² = c², where a and b are the lengths of the two shorter sides (legs) of a right triangle, and c is the length of the longest side (the hypotenuse).
To find the hypotenuse c, square the lengths of the legs a and b, add the squares together, and take the square root of the sum: c = √(a² + b²).
No, the Pythagorean theorem only works on right-angled triangles (triangles where one angle is exactly 90 degrees). For oblique triangles, you must use the Law of Sines or the Law of Cosines.
Pythagorean triples are sets of three positive integers (a, b, c) that perfectly satisfy the formula a² + b² = c². The most common primitive triples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25).
To check if three sides form a right triangle, square all three side lengths. Add the squares of the two smaller sides. If their sum equals the square of the largest side, the triangle contains a right angle.
A common real-life example is leaning a ladder against a wall. The ladder is the hypotenuse, the ground distance to the wall is leg a, and the height on the wall is leg b. Squaring ground distance and wall height gives the square of the ladder's length.
A primitive Pythagorean triple consists of three positive integers that share no common factors other than 1, like (3, 4, 5). A non-primitive triple is a multiple of a primitive one, such as (6, 8, 10), where all sides are multiplied by 2.
While named after the ancient Greek philosopher Pythagoras, the relationship between right triangle sides was known much earlier. Ancient Babylonian, Egyptian, Chinese, and Indian mathematicians documented these triples centuries before Pythagoras lived.