The Pythagorean theorem calculator finds an unknown side of a right triangle when the required side lengths are known using the fundamental geometric equation a² + b² = c².
Calculate the hypotenuse or a missing leg instantly when the required side lengths are known with step-by-step mathematical substitution, visual SVG diagrams, and unit conversions.
Select your calculation mode below, input known side lengths, and view instant results with dynamic SVG diagrams.
The Pythagorean theorem calculator provides an interactive right triangle solver for finding any unknown side when compatible measurements are known:
When both legs of a right triangle are known, the calculator can determine the hypotenuse.
When the hypotenuse and one leg are known, the calculator can determine the remaining leg.
Use the Pythagorean relationship to solve an unknown side when two compatible measurements are known.
The Pythagorean theorem describes the fundamental geometric relationship between the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two legs:
Where a and b are the leg lengths, and c is the hypotenuse.
The theorem applies specifically to right triangles.
A right triangle consists of three distinct sides defined relative to the 90-degree right angle:
One of the two perpendicular sides forming the right angle.
The second perpendicular side forming the right angle.
The side opposite the 90° angle and the longest side of the right triangle.
Leg a ⊥ Leg b | Hypotenuse c = Longest Side (Opposite 90° Angle)
To use the Pythagorean equation accurately, ensure your geometric problem satisfies these requirements:
Follow these quick steps to calculate any unknown right triangle side length:
Here is a step-by-step calculation for a right triangle with leg lengths a = 3 and b = 4:
Given: a = 3, b = 4
Formula: c = √(a² + b²)
Substitution: c = √(3² + 4²)
Calculation: c = √(9 + 16) = √25 = 5
Conclusion: The hypotenuse is 5 units.
If three side lengths are known, identify the longest side as c and check whether they satisfy the Pythagorean relationship:
Example: Test side lengths 7, 24, and 25:
7² + 24² = 25²49 + 576 = 625625 = 625Therefore, the relationship is satisfied and the triangle contains a 90° right angle.
Verify integer triples using our Pythagorean Triple Calculator.
A Pythagorean triple consists of three positive integers that satisfy a² + b² = c². Common fundamental triples include:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 40 | 41 |
When the hypotenuse and one leg are known, the Pythagorean equation can be rearranged to find the unknown leg:
This reverse formula isolates leg length a by subtracting the square of known leg b from squared hypotenuse c.
The Pythagorean theorem is essential across architecture, carpentry, geometry, and physical layout tasks:
Checking right angles, diagonal dimensions, braces, and layout measurements for foundations and wall framing.
Construction Calculator →Finding the straight-line length between vertical height and horizontal distance for safety placement.
Finding the diagonal distance across a rectangular room, wall, or floor framing.
Finding ramp length from rise and horizontal run dimensions.
Calculating straight-line distances between horizontal and vertical points for guy-wires and supports.
The Pythagorean relationship can be extended to calculate a three-dimensional diagonal or distance across spatial dimensions:
Common 3D applications include calculating 3D distance, cuboid diagonals, box diagonals, and three-dimensional coordinate spaces.
When both legs of a right triangle are equal (a = b), the equation simplifies to:
This describes the special 45-45-90 right triangle relationship where the hypotenuse is exactly √2 times the length of either leg.
Keep these practical guidelines in mind when inputting measurements into the calculator:
Avoid these frequent pitfalls when solving right triangle side lengths:
The hypotenuse is opposite the 90° angle. Never substitute a leg length in place of side c.
Use a² and b², not a + b. The sum of leg lengths is always strictly greater than hypotenuse c.
The standard equation applies strictly to right triangles. Use the Law of Cosines for acute/obtuse triangles.
Convert measurements to compatible units (e.g., converting inches to feet) before calculating.
The hypotenuse must be longer than either leg. The calculator requires c > a and c > b.
Core mathematical vocabulary associated with right-triangle side calculations:
The Pythagorean theorem calculator finds an unknown side length (hypotenuse or missing leg) of a right triangle when two required side lengths are known.
Yes, select "Find Hypotenuse (c)", enter legs a and b, and the calculator computes c = √(a² + b²).
Yes, select "Find Side (a)" or "Find Side (b)", input hypotenuse c and the known leg, and the calculator computes the missing leg using a = √(c² - b²).
No. The standard Pythagorean theorem applies strictly to right triangles containing a 90° angle. Non-right triangles require the Law of Cosines.
The hypotenuse (c) is the side opposite the 90° right angle and is always the longest side of a right-angled triangle.
Yes, our calculator supports all positive decimal values and fractional inputs.
You can select units such as centimeters (cm), meters (m), millimeters (mm), inches (in), feet (ft), or yards (yd).
The calculator displays an error message if the hypotenuse is less than or equal to a leg, as a right triangle cannot be formed.
Reverse calculation rearranges the formula to solve for an unknown leg, while the converse tests whether three given side lengths satisfy a² + b² = c² to form a right angle.
A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the equation a² + b² = c², such as (3, 4, 5).
Explore specialized right triangle tools tailored to your exact mathematical or engineering requirements: