How do I check if a triangle is right?

Square the two shorter sides, add them, and compare to the square of the longest side. If equal, it's a right triangle.

Math

Pythagorean Theorem Calculator

Calculate any missing side of a right triangle using a² + b² = c². Enter two sides and find the third. Also check if three sides form a right triangle.

Quick Answer

The Pythagorean theorem states a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle) and a, b are the two legs. To find c: c = √(a² + b²). To find a leg: a = √(c² - b²).

Find a Missing Side

Enter any 2 of the 3 sides. Leave the unknown side blank.

a² + b² = c²

Side a (leg)

Side b (leg)

Side c (hypotenuse)

Solution

Side a

Side b

Hypotenuse c (solved)

Area

½ × a × b

Perimeter

a + b + c

Step-by-Step Solution

Step 1: Identify known values

a = 3, b = 4, c = ?

Step 2: Apply the formula

c² = a² + b² = 3² + 4² = 9 + 16 = 25

Step 3: Take the square root

c = √25 = 5

Right Triangle Diagram

Right Triangle Checker

Enter all three sides to verify if a triangle is a right triangle.

Side 1

Side 2

Side 3

Yes, this is a right triangle!

3² + 4² = 25 = 5² = 25

About This Tool

The Pythagorean Theorem Calculator is a dedicated tool for solving right triangles using one of the most famous formulas in mathematics: a² + b² = c². Enter any two sides of a right triangle and instantly find the third side, along with the area and perimeter. It also includes a right triangle checker that tests whether three given sides form a right triangle. This tool is essential for geometry students, construction professionals, and anyone working with right angles.

The Pythagorean Theorem Explained

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Written algebraically: a² + b² = c², where c is always the hypotenuse. This theorem is attributed to Pythagoras of Samos (circa 570-495 BC), though evidence suggests Babylonian mathematicians knew this relationship over a thousand years earlier. It is arguably the most widely known and applied theorem in all of mathematics.

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous triple is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a Pythagorean triple is also a triple: for example, (6, 8, 10) is 2 times (3, 4, 5). Pythagorean triples are useful in construction for verifying right angles without measuring instruments, a technique known as the "3-4-5 rule."

Proofs of the Pythagorean Theorem

There are over 370 known proofs of the Pythagorean theorem, making it one of the most-proved theorems in mathematics. Euclid provided a geometric proof using area comparisons. The most elegant proof uses similar triangles formed by dropping an altitude from the right angle to the hypotenuse. U.S. President James Garfield published an original proof using a trapezoid in 1876. The abundance of proofs reflects the theorem's deep connection to fundamental mathematical concepts.

Applications in Real Life

The Pythagorean theorem is used constantly in construction (ensuring walls are square), navigation (calculating straight-line distances), architecture (diagonal bracing), computer graphics (distance between points), and physics (vector magnitudes). When a GPS calculates the distance between two points, it uses a form of the Pythagorean theorem. Carpenters use the 3-4-5 rule to check right angles. Pilots use it to calculate descent rates. The theorem extends to three dimensions as d² = x² + y² + z², making it the foundation of distance calculations in 3D space.

The Converse and Extensions

The converse of the Pythagorean theorem is equally useful: if a² + b² = c² for a triangle with sides a, b, c (where c is the largest), then the triangle is a right triangle. If a² + b² is greater than c², the triangle is acute. If a² + b² is less than c², the triangle is obtuse. This checker feature above uses the converse to test any three sides. The theorem also generalizes to the law of cosines, which handles non-right triangles.

Frequently Asked Questions

Which side is the hypotenuse?

The hypotenuse is always the longest side of a right triangle and is located opposite the 90-degree angle. In the formula a^2 + b^2 = c^2, 'c' represents the hypotenuse. If you enter two legs, the calculator finds the hypotenuse. If you enter the hypotenuse and one leg, it finds the other leg.

Can the Pythagorean theorem be used for non-right triangles?

No, the Pythagorean theorem only applies to right triangles (those with one 90-degree angle). For non-right triangles, use the law of cosines: c^2 = a^2 + b^2 - 2ab*cos(C). When the angle C is 90 degrees, cos(90) = 0, and the formula reduces to the Pythagorean theorem.

What is a Pythagorean triple?

A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem. Common examples include (3,4,5), (5,12,13), (8,15,17), and (7,24,25). Any scalar multiple of a triple is also a triple, so (6,8,10), (9,12,15), etc. are all valid. These are useful for quick mental math and construction verification.

How do I check if a triangle is a right triangle?

Enter all three side lengths into the Right Triangle Checker section above. The calculator sorts the sides, squares the two smaller ones, adds them together, and compares to the square of the largest side. If they are equal (within floating-point precision), the triangle is a right triangle.

Does this work in 3D?

The basic theorem works for 2D right triangles. In 3D, the distance formula extends to d = sqrt(x^2 + y^2 + z^2), which is essentially applying the Pythagorean theorem twice. For 3D diagonal calculations, compute the 2D diagonal first, then use it with the third dimension.

What if I get a decimal answer?

Most combinations of side lengths produce irrational results. For example, a right triangle with legs 1 and 1 has a hypotenuse of sqrt(2), which is approximately 1.414214. Only Pythagorean triples (and their multiples) produce integer results. Decimal answers are perfectly valid and normal in real-world applications.

Tax & Finance

Was this tool helpful?