Pythagorean Theorem Calculator
This free Pythagorean theorem calculator lets you find the hypotenuse or solve for a missing leg on any right triangle in seconds. Enter your two known side lengths below to generate instant, step-by-step mathematical solutions.
What is the Pythagorean Theorem?
The Pythagorean theorem is a geometric rule stating that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two legs. Expressed as a² + b² = c², it solves unknown triangle lengths across geometry, engineering, and construction.
Named after the ancient Greek mathematician Pythagoras, the relationship applies to all right-angled planar triangles. Knowing any two measurements allows you to determine the third side without measuring angles.
The Formula: a² + b² = c²
To use this relationship, the triangle must feature a 90-degree right angle. The mathematical formula is:
a² + b² = c²Variables a and b represent the shorter legs perpendicular to each other, while c represents the hypotenuse—the longest side sitting directly opposite the right angle.
How to Find the Hypotenuse and Missing Sides (With Steps)
Whether you are solving a classroom assignment or using this as a find missing side of right triangle calculator on a job site, here are the step-by-step algebraic steps:
1. How to Find the Hypotenuse with Two Sides
When you know both legs (a and b), find the hypotenuse (c) by squaring each number, adding them together, and finding the square root:
c = √(a² + b²)Example: For legs measuring a = 6 and b = 8:
1. Square both legs: 6² = 36 and 8² = 64
2. Add the results: 36 + 64 = 100
3. Take the square root: c = √100 = 10
2. How to Find a Missing Leg (a or b)
When you know the hypotenuse (c) and one leg, subtract the squared known leg from the squared hypotenuse, then calculate the square root:
a = √(c² - b²)b = √(c² - a²)Example: For hypotenuse c = 13 and leg a = 5:
1. Square both values: 13² = 169 and 5² = 25
2. Subtract the leg from the hypotenuse: 169 - 25 = 144
3. Take the square root: b = √144 = 12
Real-World Framing & Layout Applications
Right-triangle geometry serves everyday practical functions across design, trades, and site layout:
- Diagonal Bracing Calculator for Framing: Carpenters calculate diagonal wall braces and rafter lengths using height (rise) and span (run) as legs a and b to cut exact structural supports.
- Squaring Layouts with the 3-4-5 Rule: Builders mark 3 feet on one line and 4 feet on another. If the cross diagonal measures exactly 5 feet, the corner sits at a true 90-degree right angle.
- Surveying and Navigation: Surveyors calculate straight-line distances across rough elevations without running physical tapes through obstacles.
Frequently Asked Questions
What is the formula for the Pythagorean theorem?
The formula for the Pythagorean theorem is a² + b² = c². In this equation, a and b represent the two shorter legs perpendicular to each other, and c represents the hypotenuse, which is the longest side opposite the 90-degree right angle.
How do you find the missing side of a right triangle?
To find the hypotenuse (c), add the squares of legs a and b, then calculate the square root: c = √(a² + b²). To find a missing leg (a or b), subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² - b²).
Can the Pythagorean theorem be used on any triangle?
No. The Pythagorean theorem works strictly on right-angled triangles with one 90-degree angle. For acute or obtuse triangles without a right angle, you must use the Law of Cosines instead.
What is the 3-4-5 rule in construction?
The 3-4-5 rule is a practical method based on the Pythagorean theorem to verify square 90-degree corners during layout and framing. Measure 3 units along one wall, 4 units along the adjacent perpendicular wall, and check the diagonal distance between the marks. If the diagonal measures exactly 5 units (3² + 4² = 9 + 16 = 25 = 5²), the corner is perfectly square.
Pythagorean Calculator
1. Pythagorean Formula: c² = a² + b²
2. Substitute values: c² = 3² + 4² = 9 + 16 = 25
3. Take the square root: c = √25 = 5.0000
* In any right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs: c² = a² + b².
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