Common Pythagorean triples
A Pythagorean triple is a set of three positive integers that exactly satisfy a² + b² = c². The table below lists the smallest and most frequently used triples.
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 40 | 41 |
| 20 | 21 | 29 |
- Any integer multiple of a Pythagorean triple is also a valid triple — for example 6-8-10 and 9-12-15 are both multiples of 3-4-5.
- The Pythagorean theorem only holds exactly for right triangles on a flat (Euclidean) plane; it does not apply to triangles on a curved surface such as a sphere.
What is the Pythagorean theorem?
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 other two sides, called legs: a² + b² = c². It is one of the most widely used relationships in geometry, attributed to the ancient Greek mathematician Pythagoras and proven independently in multiple ancient mathematical traditions.
The theorem applies exclusively to right triangles — triangles containing exactly one 90° angle. It is used to find an unknown side when the other two are known, and underlies the distance formula, trigonometric identities and countless practical measurement problems in construction, navigation and engineering.
How to use this Pythagorean theorem calculator
- Choose what to solve for: the hypotenuse (given both legs) or a missing leg (given the hypotenuse and one leg).
- In hypotenuse mode, enter leg a and leg b.
- In missing-leg mode, enter the known leg (a) and the hypotenuse (c); the hypotenuse must be greater than the known leg.
- Read the missing side, plus the triangle's area and perimeter, calculated from all three sides.
The Pythagorean theorem formula
The same base equation rearranges depending on which side is unknown.
Common mistakes
- Applying the formula to a triangle that is not a right triangle — the theorem only holds when one angle is exactly 90°.
- Adding the hypotenuse into a² + b² = c² as if it were a leg, instead of solving c² − a² = b² for a missing leg.
- Entering a leg value equal to or larger than the hypotenuse in "missing leg" mode, which is geometrically impossible for a right triangle.
- Forgetting to take the square root after summing (or subtracting) the squares — the theorem gives the squared value first.
자주 묻는 질문
What is the Pythagorean theorem formula?
a² + b² = c², where a and b are the two legs of a right triangle and c is the hypotenuse, the side opposite the right angle.
How do you find the hypotenuse of a right triangle?
Take the square root of the sum of the squares of the two legs: c = √(a² + b²). For legs of 3 and 4, c = √(9 + 16) = √25 = 5.
How do you find a missing leg using the Pythagorean theorem?
Subtract the known leg's square from the hypotenuse's square, then take the square root: b = √(c² − a²). For a hypotenuse of 5 and a known leg of 3, b = √(25 − 9) = √16 = 4.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive whole numbers a, b and c that satisfy a² + b² = c² exactly, such as 3-4-5, 5-12-13 or 8-15-17.
Does the Pythagorean theorem work for all triangles?
No. It applies only to right triangles — those with exactly one 90° angle. For triangles without a right angle, the law of cosines (a generalization of the Pythagorean theorem) is used instead.
What is the area of a 3-4-5 right triangle?
The area of a right triangle equals half the product of its two legs: A = ½ × a × b. For legs 3 and 4, the area is ½ × 3 × 4 = 6 square units, with a perimeter of 3 + 4 + 5 = 12 units.
참고 자료
- Weisstein, Eric W. "Pythagorean Theorem" and "Pythagorean Triple." MathWorld — A Wolfram Web Resource.
- Standard geometry textbook conventions (e.g. Larson, Geometry, Cengage Learning).