Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²).
About This Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Use the calculator above for instant results.
Understanding the Pythagorean Theorem
For a right triangle, a²+b²=c², where c is the hypotenuse opposite the 90° angle and a and b are the legs. If both legs are known, c=√(a²+b²). If the hypotenuse and one leg are known, the other leg is found by rearranging: a=√(c²−b²) or b=√(c²−a²).
The theorem is a distance relationship. On a coordinate plane, the horizontal and vertical changes between two points form perpendicular legs, which leads directly to the distance formula d=√[(Δx)²+(Δy)²]. It is also widely used to compute diagonals of rectangles, verify square corners, and relate components of perpendicular vectors.
A useful reasonableness check is that the hypotenuse must be longer than either leg but shorter than the sum of the two legs. When solving for a leg, c²−a² or c²−b² must be positive. If it is zero or negative, the supplied lengths cannot form the requested nondegenerate right triangle.
How to Check a Right-Triangle Result
Substitute the completed side lengths back into a²+b²=c². For a 7-24-25 triangle, 7²+24²=49+576=625=25². This direct substitution checks both the algebra and the square-root step. A second check is ordering: c must be the largest side because it is opposite the right angle.
The theorem also supports the converse test. If three positive side lengths satisfy a²+b²=c² with c chosen as the longest side, then the triangle is right. If a²+b²>c² the angle opposite c is acute; if a²+b²<c² it is obtuse. The calculator itself is designed for the right-triangle equality, but this comparison helps interpret measured lengths. Keep every side in the same unit before squaring; unit conversion should happen before applying the theorem.
Worked Examples
- √(9+16) = √25 = 5 | Classic 3-4-5!
Answer: c = 5
- √(169-25) = √144 = 12
Answer: b = 12
- √(36+64) = √100 = 10
Answer: c = 10
- c=√(64+225)=√289=17
Answer: c = 17
- a=√(625−576)=√49=7
Answer: a = 7
Who Uses This Calculator?
Solve right triangle problems.
Calculate diagonal distances.
Right angle and distance calculations.
Demo the Pythagorean theorem.
Common Mistakes to Avoid
❌ Only works for right triangles
The Pythagorean theorem requires a 90° angle. For other triangles, use the Law of Cosines.
❌ Hypotenuse must be longest
c (hypotenuse) must be larger than either a or b. If c < a or c < b, something is wrong.
❌ Subtracting when finding the hypotenuse
When both legs are known, add their squares. Subtraction is used only after rearranging the theorem to solve for a missing leg.
❌ Forgetting to square units
If side lengths are in meters, the intermediate squared terms are in m², and the final square root returns the answer to meters.
The hypotenuse must always be the longest side, which provides another immediate geometry check.
Frequently Asked Questions
Pythagorean theorem?
a² + b² = c². Where c is the hypotenuse (side opposite the right angle) and a, b are the two legs.
Pythagorean triples?
Integer sets where a²+b²=c²: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Infinitely many exist.
Who discovered it?
Attributed to Pythagoras (~500 BC) but known to Babylonians 1,000 years earlier.
How is the distance formula related?
Horizontal and vertical coordinate differences are perpendicular legs, so the Pythagorean theorem gives d=√[(Δx)²+(Δy)²].
Can a right triangle have decimal side lengths?
Yes. Pythagorean triples are convenient integer examples, but the theorem applies to positive real side lengths.
How can I check a missing-leg result?
Square both legs and add them. The total should equal c² within any rounding used.
What should I use for a non-right triangle?
Use the Law of Cosines when the included angle is not 90° or when the triangle is not known to be right.
Formula Explorer connections
Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.