Pythagorean Theorem Calculator
Solve for the hypotenuse, a missing leg, or the area of any right triangle in seconds.
Right Triangle Calculator
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Pythagorean Triples Reference Table
This reference table lists common Pythagorean triples, which are sets of whole numbers that satisfy the Pythagorean theorem exactly.
| Side a | Side b | Hypotenuse c | Check: a squared + b squared |
|---|---|---|---|
| 3 | 4 | 5 | 9 + 16 = 25 |
| 5 | 12 | 13 | 25 + 144 = 169 |
| 8 | 15 | 17 | 64 + 225 = 289 |
| 7 | 24 | 25 | 49 + 576 = 625 |
| 20 | 21 | 29 | 400 + 441 = 841 |
| 9 | 40 | 41 | 81 + 1600 = 1681 |
Pythagorean Theorem Explained
The Pythagorean theorem explains the relationship between the three sides of a right triangle. It states that the square of the hypotenuse equals the sum of the squares of the other two sides. This rule applies only to right triangles, which are triangles that contain one angle of exactly 90 degrees.
The two shorter sides of a right triangle are called legs, often labeled a and b, while the longest side opposite the right angle is called the hypotenuse, labeled c.
How to Find the Hypotenuse
How to find the hypotenuse starts with squaring both leg lengths and adding them together. Once you have that sum, take the square root to get the length of c.
How to Find a Missing Leg
How to find a missing leg requires subtracting the square of the known leg from the square of the hypotenuse, then taking the square root of that difference.
How to Calculate the Area of a Right Triangle
How to calculate the area of a right triangle uses the two legs as base and height. Multiply leg a by leg b and divide the result by 2 to get the area.
Real World Uses for the Pythagorean Theorem
Real world uses for the Pythagorean theorem appear in construction, navigation, engineering, and everyday measuring tasks. Builders use it to check that corners are square, while electricians use it to calculate conduit lengths across diagonal runs.
Construction and Carpentry
Construction and carpentry rely on the 3 4 5 rule, a Pythagorean triple, to confirm that walls, foundations, and frames meet at perfect right angles without needing a protractor.
Navigation and Distance
Navigation and distance calculations use the theorem to find the shortest straight line path between two points when horizontal and vertical distances are known, such as plotting a direct route across a grid.
Screen and Display Sizing
Screen and display sizing depends on the theorem because a screen’s diagonal measurement, width, and height form a right triangle, allowing manufacturers to calculate diagonal size from width and height.
Frequently Asked Questions
The Pythagorean theorem formula is a squared plus b squared equals c squared, where a and b are the two legs of a right triangle and c is the hypotenuse. This formula lets you find any missing side if you know the other two.
To find the hypotenuse, square the lengths of both legs, add the results together, then take the square root of that sum. The formula is c equals the square root of a squared plus b squared.
To find a missing leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root of the result. For example, side a equals the square root of c squared minus b squared.
A Pythagorean triple is a set of three positive whole numbers a, b, and c that satisfy the equation a squared plus b squared equals c squared. The most common example is 3, 4, and 5, since 9 plus 16 equals 25.
No, the Pythagorean theorem only works for right triangles, which contain one 90 degree angle. For other triangles, the law of cosines or law of sines must be used instead.
The area of a right triangle equals one half multiplied by the length of leg a multiplied by the length of leg b. This is because the two legs form the base and height of the triangle.