What is the Pythagorean theorem?
The Pythagorean theorem describes the relationship between the three sides of a right triangle. If a and b are the perpendicular legs and c is the hypotenuse, then the sum of the squares of the legs equals the square of the hypotenuse.
The theorem only applies directly to right triangles. The hypotenuse is always the side opposite the 90° angle, and it is therefore the longest side. This distinction matters: if you put the wrong side in the role of c, the calculation will not represent the geometry correctly.
Key idea: “Right triangle” is not optional here. A triangle with no 90° angle does not satisfy a² + b² = c² in this direct form. For general triangles, the law of cosines is the broader relationship.
Pythagorean theorem formulas
The same equation can be rearranged depending on which side is missing. Keeping the algebra visible makes it easier to understand what the calculator is doing.
- a
- first leg
- b
- second leg
- c
- hypotenuse, opposite 90°
Pythagorean theorem examples
Working through the algebra step by step is useful because it reveals why the calculator produces the number, not just what the number is.
Find the hypotenuse
The familiar 3–4–5 triangle is the smallest positive integer Pythagorean triple.
c = 13, a = 5 → find b
This gives the 5–12–13 Pythagorean triple.
a = 7, b = 9 → find c
Not every right triangle has integer side lengths; the theorem works equally well with decimals.
a = 8, b = 15
The same two legs determine the area immediately once their product is known.
Pythagorean triples
A Pythagorean triple is a set of positive integers (a, b, c) that satisfies a² + b² = c². They are useful for recognizing exact right-triangle dimensions without a calculator.
| Triple | Check | Meaning |
|---|---|---|
| 3, 4, 5 | 9 + 16 = 25 | Smallest positive integer triple |
| 5, 12, 13 | 25 + 144 = 169 | Common construction and geometry example |
| 8, 15, 17 | 64 + 225 = 289 | Another exact integer right triangle |
| 7, 24, 25 | 49 + 576 = 625 | Useful when a larger leg is involved |
Multiplying every member of a triple by the same positive number creates another valid right triangle. For example, 3–4–5 scaled by 2 becomes 6–8–10.
Where the Pythagorean theorem is used
The theorem appears whenever two perpendicular distances form a right triangle and the straight-line distance is needed.
📐 Construction
A 3–4–5 triangle can establish a reliable right angle on a site. Longer scaled triples work the same way.
🪜 Ladders
A wall, ground distance, and ladder form a right triangle when the wall is vertical and the ground is level.
🖥️ Diagonals
The diagonal of a rectangle with side lengths a and b is √(a²+b²).
🗺️ Coordinate geometry
The distance formula is a direct application of the theorem to horizontal and vertical differences.
Coordinate connection: For two points (x₁,y₁) and (x₂,y₂), the distance is √[(x₂−x₁)² + (y₂−y₁)²]. The horizontal and vertical differences form the two legs of a right triangle.
Pythagorean theorem FAQ
No. The direct equation a² + b² = c² applies to right triangles. For a general triangle, the law of cosines is the broader relationship.
It is the side opposite the 90° angle, and it is always the longest side of a right triangle.
Yes. Enter c and either a or b. The calculator rearranges the equation and takes the square root of the remaining difference.
Because c² = a² + b², and both a² and b² are positive. Therefore c² is greater than either leg square, so c is greater than either leg.
Those values cannot form a real right triangle with a as a leg and c as the hypotenuse. The calculator flags the input instead of returning a non-real length.