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Easily find the length of the hypotenuse of a right-angled triangle given the lengths of the other two sides.
5.0000
Everything you need to know
The Pythagorean theorem is one of the most famous and useful principles in all of mathematics. Discovered by the ancient Greek mathematician Pythagoras around 500 BCE, it describes the fundamental relationship between the three sides of a right triangle. Our calculator makes it effortless to find any missing side when you know the other two.
Whether you are a student learning geometry, a carpenter framing a house, a sailor navigating open waters, or a DIYer building a deck, the Pythagorean theorem provides a reliable way to calculate distances and ensure square corners.
For any right triangle (a triangle with one 90-degree angle):
a² + b² = c²
Where:
If you know both legs, rearrange the formula: c = √(a² + b²)
Example: A right triangle has legs of 3 feet and 4 feet. c = √(3² + 4²) = √(9 + 16) = √25 = 5 feet
This 3-4-5 relationship is the most famous right triangle and appears constantly in construction and design.
If you know the hypotenuse and one leg, rearrange the formula: a = √(c² - b²)
Example: A ladder leans against a wall. The base is 6 feet from the wall, and the ladder reaches 10 feet up. a = √(10² - 6²) = √(100 - 36) = √64 = 8 feet Wait — actually, if the ladder is 10 feet and the base is 6 feet, the height is: a = √(10² - 6²) = √(100 - 36) = √64 = 8 feet
One of the most elegant proofs involves arranging four identical right triangles inside a square:
This proof has been reproduced for over 2,000 years and remains one of the most beautiful demonstrations in mathematics.
Certain right triangles appear so frequently that memorizing their ratios saves time:
| Leg 1 | Leg 2 | Hypotenuse |
|---|---|---|
| 3 | 4 | 5 |
| 6 | 8 | 10 |
| 9 | 12 | 15 |
| 12 | 16 | 20 |
| 15 | 20 | 25 |
Any multiple of 3-4-5 forms a valid right triangle. This is the most common triangle in construction.
| Leg 1 | Leg 2 | Hypotenuse |
|---|---|---|
| 5 | 12 | 13 |
| 10 | 24 | 26 |
| Leg 1 | Leg 2 | Hypotenuse |
|---|---|---|
| 8 | 15 | 17 |
Both legs are equal, and the hypotenuse is leg × √1.414.
| Leg | Hypotenuse |
|---|---|
| 1 | 1.414 |
| 5 | 7.07 |
| 10 | 14.14 |
The sides are in a 1 : √3 : 2 ratio.
| Short Leg | Long Leg | Hypotenuse |
|---|---|---|
| 1 | 1.732 | 2 |
| 5 | 8.66 | 10 |
| 10 | 17.32 | 20 |
Squaring a foundation: To ensure a foundation corner is exactly 90 degrees, measure 3 feet along one wall and 4 feet along the adjacent wall. The diagonal between those points should be exactly 5 feet. If not, adjust until it is.
Roof pitch: A roof that rises 6 feet over a 12-foot run has a rafter length of: √(6² + 12²) = √(36 + 144) = √180 = 13.42 feet
Stair stringers: For stairs with a 7-inch rise and 10-inch tread, each stringer section measures: √(7² + 10²) = √(49 + 100) = √149 = 12.21 inches
Distance between two points: On a map, if Point A is 4 miles east and 3 miles north of Point B, the straight-line distance is: √(4² + 3²) = 5 miles
GPS coordinates: The theorem underlies the distance calculations used by GPS systems to determine your location.
Vector addition: When two forces act at right angles, their combined magnitude is found using the Pythagorean theorem.
Optics: The theorem calculates the path length of light through lenses and prisms.
Computer graphics: 3D rendering engines use the theorem constantly to calculate distances, angles, and lighting.
TV screen size: A 65-inch TV with a 16:9 aspect ratio has width and height calculable using the theorem. The diagonal is 65 inches, and the ratio of width to height is 16:9.
Ladder safety: OSHA recommends placing a ladder base one foot away from the wall for every four feet of height. The theorem tells you the exact ladder length needed.
Painting a diagonal wall: To cut a diagonal piece of trim or wallpaper, calculate the diagonal length using the wall's width and height.
Enter the two known leg lengths (a and b). The calculator computes c = √(a² + b²).
Enter the hypotenuse and one known leg. The calculator computes the missing leg.
Enter all three sides. If a² + b² = c², the triangle is a right triangle.
The hypotenuse is always the longest side and is always opposite the right angle. Never use a leg length as the hypotenuse unless it truly is the longest side.
The theorem uses a² and b², not just a and b. Squaring is essential.
The Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines: c² = a² + b² - 2ab cos(C).
When working through multi-step problems, carry full precision until the final answer. Rounding intermediate values introduces errors.
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A right triangle has legs of 5 feet and 12 feet. Find the hypotenuse.
Solution:
A right triangle has a hypotenuse of 15 inches and one leg of 9 inches. Find the missing leg.
Solution:
For any right triangle with legs a and b and hypotenuse c:
a² + b² = c²
Finding the Hypotenuse: c = √(a² + b²)
Finding a Missing Leg: a = √(c² - b²) or b = √(c² - a²)
Example: For a right triangle with legs 3 and 4: c = √(3² + 4²) = √(9 + 16) = √25 = 5
While named after Pythagoras, evidence suggests the theorem was known to Babylonian mathematicians over 1,000 years before Pythagoras. The ancient Egyptians also used 3-4-5 triangles to lay out square corners for construction.
Yes. The three-dimensional extension is: a² + b² + c² = d², where d is the space diagonal of a rectangular prism.
Use the Law of Cosines: c² = a² + b² - 2ab cos(C). When angle C is 90 degrees, cos(90°) = 0, and the formula reduces to the Pythagorean theorem.
Yes, infinitely many. A Pythagorean triple is any set of three whole numbers that satisfy a² + b² = c². Examples include 5-12-13, 8-15-17, 7-24-25, and 20-21-29.
The calculator uses standard floating-point arithmetic and is accurate to many decimal places. For construction, round to the nearest 1/16 inch or millimeter.
This calculator is provided for informational and educational purposes only. Results are calculated based on standard mathematical formulas and your inputs. While we strive for accuracy, we do not guarantee that results are error-free. Always verify important calculations independently. Users are responsible for the accuracy of their inputs and should consult appropriate references or professionals for critical applications. We are not liable for any decisions made based on these calculations.
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