📐Pythagorean Theorem Calculator

The Pythagorean theorem calculator makes it fast and simple to find any missing side of a right angle triangle. Whether you need to find the hypotenuse or solve for one of the shorter legs, just enter the two known sides and the calculator does the rest. The theorem, expressed as c squared equals a squared plus b squared, is one of the most widely applied formulas in geometry, construction, navigation, and physics.

How to Use the Pythagorean Theorem to Find the Hypotenuse

The hypotenuse is the longest side of a right triangle. It sits directly opposite the 90-degree angle. To find it when you know both legs, the formula is straightforward:

• Square leg a to get a squared.
• Square leg b to get b squared.
• Add the two results together.
• Take the square root of the sum. The result is c, the hypotenuse.

For example, a right triangle with legs of 6 and 8 has a hypotenuse of sqrt(36 + 64) = sqrt(100) = 10. This is one of the most well-known Pythagorean triples in geometry. Carpenters use this exact method, called the 3-4-5 rule, to check that corners are perfectly square during construction.

Right Triangle Side Calculator with a Squared Plus b Squared

The relationship a squared plus b squared equals c squared can be rearranged to solve for either leg when the hypotenuse is known. If you know the hypotenuse c and one leg b, the missing leg a equals the square root of (c squared minus b squared).

A practical example: a ladder 13 feet long leans against a wall, with its base placed 5 feet from the wall. The height it reaches is sqrt(169 - 25) = sqrt(144) = 12 feet. This kind of problem appears in construction, firefighting, and general home improvement.

Select which side you are solving for at the top of the calculator, enter the two known values, and the missing side is calculated instantly along with the triangle's area, perimeter, and angles.

Pythagorean Theorem Calculator with Steps

Beyond finding the missing side, this calculator also returns the triangle's interior angles, area, and perimeter.

Angles are found using inverse trigonometric functions. Angle alpha, which is opposite leg a, equals arcsin(a divided by c). Angle beta equals arcsin(b divided by c). Because one angle is always 90 degrees in a right triangle, the other two must sum to 90 degrees as well.

The area of a right triangle is (1/2) multiplied by leg a multiplied by leg b. Since the two legs are perpendicular, they act as the base and height directly, making this calculation simpler than for other triangle types.

The Distance Formula and Real-World Applications

The distance formula used in coordinate geometry is a direct extension of Pythagoras. The straight-line distance between two points (x1, y1) and (x2, y2) equals the square root of ((x2 minus x1) squared plus (y2 minus y1) squared). This is the hypotenuse of a right triangle formed by the horizontal and vertical distances between the two points.

Real-world uses of the Pythagorean theorem include:

• Construction: Calculating rafter lengths, diagonal bracing, and squaring foundations.
• Navigation: Finding straight-line distances when traveling known east-west and north-south distances.
• Screen sizes: A television's listed size is its diagonal measurement, calculated from its width and height using the theorem.
• Physics: Adding perpendicular force vectors and finding resultant magnitudes.

Common Pythagorean Triples

Pythagorean triples are sets of three whole numbers that satisfy a squared plus b squared equals c squared exactly, with no rounding required. Knowing a few of these saves time when working with right triangles in whole-number problems.

• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25
• 9, 40, 41

Any whole-number multiple of a Pythagorean triple also forms a valid right triangle. Doubling the 3-4-5 triple gives 6-8-10, which satisfies the same relationship.