Pythagorean Theorem Calculator: Find Hypotenuse or Leg of a Right Triangle
Free Pythagorean theorem calculator: find the missing side of a right triangle. Solve for hypotenuse (c) or leg (a or b) using a² + b² = c². How to find the hypotenuse, how to find a leg, step-by-step solver, perimeter, area, and angles. Used in geometry, construction, and surveying.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:The missing side (a, b, or c), a step-by-step logic trace showing the substitution into a² + b² = c² and the intermediate squares (e.g. 9 + 16 = 25) before the square root, a dynamic right triangle diagram that scales to your inputs and highlights the calculated side, plus perimeter, area, and the two non-right angles in degrees.
- Ideal Users:Students & geometry: Find hypotenuse or leg, check homework, see step-by-step. Teachers: Demonstrate the theorem with a visual triangle and algebraic steps. Construction & DIY: Diagonal length of a rectangle, rafter length, right-angle layout. Surveying & carpentry: Verify 3-4-5 (or other) right triangles. Anyone: Quick right triangle solver for two known sides.
- Scope & Limits:Right triangles only. All sides must be positive. The hypotenuse must be the longest side; otherwise the calculator shows a mathematical error and asks you to correct the inputs.
This Pythagorean theorem calculator solves for the missing side of a right triangle: enter any two of side a, side b, or hypotenuse c, and the third is computed instantly. It is commonly used in high school geometry and in construction and surveying for diagonal lengths and right-angle checks.
What Is the Pythagorean Theorem? Definition and Formula
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 two legs. The formula is:
a² + b² = c²
where c is the hypotenuse and a and b are the legs. From this you can find the hypotenuse: c = √(a² + b²)
or find a leg: a = √(c² − b²)
or b = √(c² − a²)
This Pythagorean theorem calculator applies these formulas automatically when you enter two sides.How to Find the Hypotenuse of a Right Triangle
To find the hypotenuse when you know both legs: add the squares of the two legs and take the square root. So c = √(a² + b²). For example, if a = 3 and b = 4, then a² + b² = 9 + 16 = 25, and c = √25 = 5. In the calculator, enter Side a and Side b; the hypotenuse c is computed and the Logic Trace shows the intermediate squares and final square root.
How to Find a Leg When You Have the Hypotenuse and the Other Leg
To find one leg when you have the hypotenuse and the other leg, rearrange the theorem: a² = c² − b² or b² = c² − a², then take the square root. For a valid right triangle, the hypotenuse must be longer than each leg, so c² − a² and c² − b² are positive. Enter the hypotenuse and one leg in the calculator; the other leg is computed and highlighted on the triangle diagram.
Why the Hypotenuse Must Be the Longest Side
In a right triangle, the hypotenuse is always the longest side because it lies opposite the 90° angle. If a ≥ c or b ≥ c, the triangle is invalid and the Pythagorean relation cannot hold. The calculator displays a mathematical error in that case and asks you to ensure the hypotenuse is greater than both legs.
Real-World Use: Construction, Surveying, and Diagonal Length
The Pythagorean theorem is widely used in construction and surveying. To find the diagonal length of a rectangle (e.g. a room or a frame), use the two sides as legs: diagonal = √(length² + width²). Carpenters use the 3-4-5 rule (and multiples like 6-8-10) to lay out right angles: if two sides are 3 and 4 units, the diagonal is 5 units. Surveyors use it for horizontal and vertical offsets. This calculator gives the same result and shows the steps, so you can use it to find the length of a triangle side in any consistent units.
Perimeter, Area, and Angles of a Right Triangle
Once the three sides are known: Perimeter = a + b + c, Area = ½ab (half the product of the two legs). The non-right angles can be found from the sides (e.g. tan A = a/b). The calculator displays perimeter, area, and angles ∠A and ∠B in degrees in the results panel.
Pythagorean Theorem Calculator FAQ
? What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs. Formula:
a² + b² = c²
where c is the hypotenuse. This calculator finds the missing side when you enter any two of a, b, and c.? How do I find the hypotenuse of a right triangle?
If you know both legs (a and b), use
c = √(a² + b²)
Enter Side a and Side b in the calculator; the hypotenuse c is computed automatically, with a step-by-step logic trace showing the squares and square root.? How do I find a leg when I have the hypotenuse and the other leg?
Use a² = c² − b² or b² = c² − a², then take the square root. Example: if c = 5 and a = 3, then b² = 25 − 9 = 16, so b = 4. Enter the hypotenuse and one leg in the calculator to get the other leg.
? Why must the hypotenuse be the longest side?
The hypotenuse is opposite the 90° angle and is always the longest side in a right triangle. If a ≥ c or b ≥ c, the triangle is invalid and the formula does not apply. The calculator shows a mathematical error in that case.
? Can I use the Pythagorean theorem for non-right triangles?
No. a² + b² = c² applies only to right triangles. For other triangles use the Law of Cosines or other methods. This tool is for right triangles only.
? What units can I use for the sides?
Use any consistent linear units (e.g. meters, feet, inches). The calculator does not convert units. Perimeter and area are in the same units (e.g. meters and m²) as the sides.