Pythagorean Theorem Calculator: Solve A² + b² = c² Step By Step

The Pythagorean theorem is one of the most used formulas in math, building and design. This guide shows how to find any side, how to check whether a triangle is right, and how to use the theorem on real jobs like ladders, TVs and floor layouts. Every number below was checked in Python.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any calculator mentioned here.

What The Pythagorean Theorem Says

The Pythagorean theorem says that in a right triangle, the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². It only works on right triangles, the ones with a 90° angle.

The parts have names:

• Legs (a and b): the two sides that meet at the right angle, in either order.
• Hypotenuse (c): the side across from the right angle, always the longest.

Picture a square on each side of a 3-4-5 triangle. The squares on the legs have areas 9 and 16, which add up to 25, the area of the square on the hypotenuse. Math is Fun's Pythagoras page has an animated version of this picture.

How To Use The Pythagorean Theorem Calculator

The calculator above solves for any side when you know the other two.

1. Choose a tab. "Find a Side" for most problems, "Is It a Right Triangle?" to test three sides, or "Distance Between Points" for coordinates.
2. Enter the two sides you know. Leave the unknown box empty. Pick your unit, including feet and inches.
3. Read the result. You get the missing side as an exact value and a decimal, the steps, both angles, the area and a drawing.

How To Find The Hypotenuse

To find the hypotenuse, square both legs, add them, then take the square root: c = √(a² + b²).

For legs of 7 and 9:

1. Square each leg: 7² = 49 and 9² = 81.
2. Add: 49 + 81 = 130.
3. Take the square root: c = √130 ≈ 11.40.

Sense check: the hypotenuse must be longer than either leg but shorter than both legs added together. 11.40 is between 9 and 16.

How To Find A Missing Leg

To find a leg, square the hypotenuse, subtract the square of the known leg, then take the square root: a = √(c² − b²).

For a hypotenuse of 13 and a leg of 5:

1. Square both: 13² = 169 and 5² = 25.
2. Subtract: 169 − 25 = 144.
3. Take the square root: a = √144 = 12.

If you add by mistake, you get √194 ≈ 13.93, longer than the hypotenuse, which is impossible.

Simplifying Square Root Answers

Teachers often want the exact simplified radical instead of a decimal. Pull the largest perfect square out from under the root.

For legs of 6 and 6, c = √72. Since 72 = 36 × 2, √72 = √36 × √2 = 6√2, which is about 8.485.

For measurements, round to what you can actually measure, like the nearest tenth of an inch. The scientific calculator handles square roots if you want to check by hand.

Pythagorean Triples To Memorize

A Pythagorean triple is a set of three whole numbers that fit a² + b² = c² exactly. Knowing the common ones lets you spot answers without a calculator.

Any multiple of a triple is also a triple. Doubling 3-4-5 gives 6-8-10, and tripling it gives 9-12-15.

To build new triples, pick whole numbers m > n and use m² − n², 2mn and m² + n². With m = 3 and n = 2, you get 5, 12 and 13.

A Babylonian clay tablet known as Plimpton 322, written around 1800 BC and now held at Columbia University, lists sets of numbers that form right triangles, according to this University of British Columbia overview.

Special Right Triangles

Two right triangles show up so often that their side ratios are worth memorizing.

The 45-45-90 ratio also works backward. If the hypotenuse is 10, each leg is 10 ÷ √2 = 5√2, or about 7.071.

How To Check If A Triangle Is A Right Triangle

The converse of the theorem lets you test any three sides. Put the longest side as c. If a² + b² equals c², the triangle is right. If not, compare the two values to see what kind of triangle it is.

First make sure the sides can form a triangle at all. The two shorter sides must add up to more than the longest. The sides 2, 3 and 6 fail, because 2 + 3 = 5 is less than 6.

Builders use this to square corners: mark 3 feet along one wall and 4 feet along the other. If the diagonal is exactly 5 feet, the corner is 90°. Larger rooms use 6-8-10 or 9-12-15.

Finding The Angles, Area And Perimeter

Once you know all three sides, you can find everything else about a right triangle. The calculator shows these automatically.

For a 3-4-5 triangle:

• Angles: use inverse tangent. The angle across from side 3 is tan⁻¹(3/4) ≈ 36.87°. The other is tan⁻¹(4/3) ≈ 53.13°. Together with 90°, they add up to 180°.
• Area: half the product of the legs: (3 × 4) ÷ 2 = 6 square units.
• Perimeter: add the sides: 3 + 4 + 5 = 12 units.
• Height to the hypotenuse: (a × b) ÷ c = 12 ÷ 5 = 2.4 units.

On a regular calculator, use degree mode. In radian mode, tan⁻¹(0.75) shows 0.6435 instead of 36.87.

Using The Theorem With Feet And Inches

Measuring jobs often mix feet and inches, and the formula needs a single unit. Convert everything to inches first, solve, then convert back.

Legs of 5 ft 6 in and 8 ft:

1. Convert: 5 ft 6 in = 66 inches, and 8 ft = 96 inches.
2. Square and add: 66² + 96² = 4,356 + 9,216 = 13,572.
3. Square root: √13,572 ≈ 116.5 inches.
4. Convert back: 116.5 ÷ 12 = 9 with 8.5 left over, so about 9 ft 8½ in.

Never write 5 ft 6 in as 5.6 ft. It is 5.5 ft, since 6 inches is half a foot. For fractional lengths like 7¾ inches, the fraction calculator can turn mixed numbers into decimals.

Real-World Uses Of The Pythagorean Theorem

The theorem solves any problem where two distances meet at a right angle.

Ladder height. A 12-foot ladder stands 4 feet from a wall. How high does it reach? The ladder is the hypotenuse: √(12² − 4²) = √128 ≈ 11.31 feet.

TV size. TVs are sold by their diagonal. A 55-inch TV with a 16:9 screen is about 47.9 inches wide and 27.0 inches tall.

Box diagonal. The theorem extends to three dimensions: d = √(l² + w² + h²). A box 3 by 4 by 12 inches has a diagonal of √(9 + 16 + 144) = √169 = 13 inches.

The Distance Formula Is The Pythagorean Theorem

The distance formula for two points on a graph is the Pythagorean theorem in disguise: d = √((x₂ − x₁)² + (y₂ − y₁)²). The horizontal and vertical gaps are the legs, and the distance is the hypotenuse.

For the points (1, 2) and (7, 10), the gaps are 6 and 8, so the distance is √(36 + 64) = √100 = 10.

Common Pythagorean Theorem Mistakes

Most errors come from using the wrong side as c or mixing up the operations.

Summary

• The Pythagorean theorem, a² + b² = c², works only on right triangles, with c as the hypotenuse.
• Find the hypotenuse with c = √(a² + b²), and a leg with a = √(c² − b²).
• Compare a² + b² with c² to tell right, acute and obtuse triangles apart.
• Convert to one unit before solving, especially with feet and inches.

Next step: enter your two known sides in the Pythagorean theorem calculator at the top of this page. For other equations, try the equation solver, or see every tool on the Calc Solver homepage.

Frequently Asked Questions