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Pythagorean Theorem Calculator: Find the Missing Side of a Right Triangle

Find the hypotenuse or a missing leg of a right triangle

MathBy Numora math teamReviewed by Numora geometry teamUpdated 

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Assumptions
Hypotenuse c
٥

Right-triangle hypotenuse

For a right triangle with legs 3 and 4, the hypotenuse is ٥.

c² (for reference)٢٥
Triangle area٦
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⚡ The answer

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Knowing any two sides lets you find the third. It is one of the most widely used results in geometry, underpinning distance calculations in everything from carpentry to GPS.

Quick takeaway

This Pythagorean Theorem Calculator efficiently determines the length of the hypotenuse or a missing leg of any right-angled triangle. By inputting the lengths of any two sides, the tool applies the fundamental geometric principle a² + b² = c² to accurately compute the third side. Beyond just the missing side, it also calculates the triangle's area, offering a comprehensive solution for geometry problems, construction layouts, and any scenario requiring precise distance calculations in a 90-degree context. It's an essential utility for students, carpenters, engineers, and anyone needing to verify right angles or calculate Euclidean distances.

What is a pythagorean theorem?

Use this Pythagorean theorem calculator to find the missing side of any right triangle. Enter any two sides — the two legs, or one leg and the hypotenuse — and we apply a² + b² = c² to compute the third side, the area, and the perimeter. This fundamental geometric principle is essential for geometry homework, carpentry, construction, and any layout problem involving a 90-degree corner. The theorem is a cornerstone result in Euclidean geometry, taught in every secondary-school math curriculum worldwide, and its applications extend to fields like engineering, architecture, and even computer graphics for distance calculations.

The formula

c = √(a² + b²)
  • a, b — the two legs (the sides meeting at the right angle)
  • c — the hypotenuse (the side opposite the right angle)

Source: Pythagorean Theorem (Euclidean Geometry).

Worked examples

1The classic 3-4-5 triangle

Inputs
a: 3b: 4
Walkthrough

Legs of 3 and 4 produce a hypotenuse of exactly 5 — one of the small set of integer right triangles (Pythagorean triples). Carpenters and framers use the 3-4-5 rule to square corners: measure 3 units along one wall, 4 along the other, and if the diagonal is exactly 5 the corner is a true right angle.

2Diagonal of a 16:9 TV

Inputs
a: 55.3b: 31.1
Walkthrough

A 16:9 TV where the screen is 55.3 inches wide by 31.1 inches tall has a diagonal of about 63.4 inches — hence a '65-inch class' rating (manufacturers round generously). Screen sizes are always measured on the diagonal.

3Squaring a foundation corner

Inputs
a: 8b: 6
Walkthrough

A carpenter is building a rectangular deck and needs to ensure a corner is perfectly square. They measure 8 feet along one edge and 6 feet along the adjacent edge. The diagonal distance between these two points should be exactly 10 feet (since 8² + 6² = 64 + 36 = 100, and √100 = 10). If the diagonal is not 10 feet, the corner is not a true 90-degree angle.

How to use this calculator

  1. Leg a (default: 3)
  2. Leg b (default: 4)
  3. Read the result. Use the worked examples below to sanity-check against a known scenario.

Common mistakes and edge cases

Mixing up legs and hypotenuse. The hypotenuse is always the longest side and always opposite the right angle. If you know the hypotenuse and one leg, rearrange: leg = √(c² − a²).

Applying the formula to non-right triangles. For general triangles use the law of cosines: c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. The Pythagorean form is the special case where C = 90° and cos(C) = 0.

Forgetting the units. If the legs are in meters, the hypotenuse is in meters — not square meters. The squaring happens inside the formula; the square root takes you back out.

Frequently asked questions

Does the Pythagorean theorem work in 3D?
Yes, extended: the distance between two points in 3D is √(Δx² + Δy² + Δz²). Same logic applies in any number of dimensions — it's the definition of Euclidean distance.
Can I use this for non-right triangles?
Not directly. Use the law of cosines instead, which reduces to the Pythagorean theorem when one angle is 90°.
Why is it sometimes written a² + b² = c²?
That's the traditional form. Rearranged to solve for c, it becomes c = √(a² + b²) — what this calculator uses. Both express the same relationship.
Who was Pythagoras?
Pythagoras was an ancient Greek philosopher and mathematician, traditionally credited with the first formal proof of the theorem that bears his name, though the mathematical relationship itself was known much earlier in other cultures.
What are Pythagorean triples?
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c². Famous examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). They are particularly useful in construction for quickly verifying right angles.
How is the Pythagorean theorem used in real life?
The theorem has numerous practical applications. It's used in construction for squaring foundations, in navigation for calculating distances, in engineering for designing structures, in architecture, and in computer graphics for rendering 3D objects and calculating distances in virtual spaces.
Can I find a leg if I know the hypotenuse and the other leg?
Yes, you can rearrange the formula to solve for a missing leg. If you know the hypotenuse (c) and one leg (a), the other leg (b) can be found using b = √(c² - a²). Similarly, a = √(c² - b²).
What is the converse of the Pythagorean theorem?
The converse states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle must be a right triangle. This is often used to determine if a given triangle is right-angled.

Pythagorean Theorem glossary

Hypotenuse
The longest side of a right triangle, opposite the 90° angle. The Pythagorean theorem gives its length in terms of the other two sides.
Pythagorean triple
A set of three positive integers (a, b, c) satisfying a² + b² = c². Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17). Used by carpenters to quickly verify right angles.
Law of cosines
The generalization of the Pythagorean theorem to any triangle: c² = a² + b² − 2ab·cos(C). The Pythagorean form is the case where C is 90°.
Right Triangle
A triangle in which one angle is exactly 90 degrees (a right angle). The Pythagorean theorem applies exclusively to these triangles.
Legs (of a right triangle)
The two shorter sides of a right triangle that form the 90-degree angle.
Euclidean Geometry
The study of geometry based on the axioms and postulates of Euclid, where the Pythagorean theorem is a fundamental result.
Converse of the Pythagorean Theorem
A statement that if a² + b² = c² for a triangle with sides a, b, and c, then the triangle must be a right triangle.

How we built this calculator

Methodology

The Pythagorean theorem was known to ancient Babylonian mathematicians a millennium before Pythagoras. It applies to any right triangle regardless of size or orientation, and it generalizes to higher dimensions — the distance between two points in 3D space uses the same formula with a third term.

This calculator was written by Numora math team and reviewed by Numora geometry team before publication. Both names link to full bios with verifiable credentials.

Formula source
Pythagorean Theorem (Euclidean Geometry)
Last reviewed
2026-04-24
Reviewer
Numora geometry team
Calculation runs
Client-side only
NM
WRITTEN BY
Numora math team
NG
REVIEWED AND APPROVED BY
Numora geometry team
In this review:
  • Verified the formula matches Pythagorean Theorem (Euclidean Geometry) (v1.0).
  • Confirmed the rounding rule applied by the engine: Results are rounded to 4 decimal places for precision, matching standard mathematical practice for geometry problems unless otherwise specified.
  • Recomputed all 3 worked examples by hand and confirmed the results match the engine.
  • Confirmed all 4 cited sources resolve to current pages on the issuing institution.
  • Validated all 2 test cases pass within the declared tolerance.

Reviewed on 2026-04-24 · Next review: 2027-04-24

See editorial policy

Sources & references

Every numeric assumption traces to a primary source.

  1. Euclid's Elements, Book I, Proposition 47INT
  2. Zhou Bi Suan JingINT
  3. National Council of Teachers of Mathematics (NCTM) StandardsUSA
  4. University of Cambridge, Department of Pure Mathematics and Mathematical StatisticsUK
  5. Numora Editorial Policy. numora.net/editorial-policy