Pythagorean Calculator
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Other tools you may find usefulPythagorean Theorem - history, meaning and geometric proofs
The Pythagorean Theorem is one of the most important and best-known theorems in geometry, describing the relationship between the sides of a right triangle. It is attributed to the Greek mathematician and philosopher Pythagoras of Samos (6th century BC), although historical evidence indicates that this relationship was known much earlier to the Babylonians, Indians and Chinese. The theorem states that in any right triangle, the sum of the areas of the squares on the adjacent sides is equal to the area of the square on the hypotenuse. It is of fundamental importance in navigation, construction, geodesy and 3D computer graphics (for calculating distances in space).
Our online Pythagorean theorem calculator allows you to quickly calculate any side of a right triangle. Just enter the lengths of two sides to immediately receive the third one, along with the written formula and the result in the form of a simplified square root.
The formula of the Pythagorean theorem
The classic formulation of the Pythagorean theorem for a triangle with sidesa i band hypotenuseclooks as follows:
a² + b² = c²
Depending on which side we want to calculate, we transform the formula to the following forms:
- Calculation of the hypotenuse (c):
c = √(a² + b²) - Calculation of the hypotenuse (a):
a = √(c² - b²) - Calculation of the hypotenuse (b):
b = √(c² - a²)
Table of common Pythagorean triples
Pythagorean triples are sets of three integers that satisfy the Pythagorean equation. The table below shows the most popular of them and the areas and perimeters of the resulting right triangles:
| Side (a) | Side (b) | Hypotenuse (c) | Perimeter of the triangle (a + b + c) | Area of the triangle (0.5 × a × b) |
|---|---|---|---|---|
| 3 | 4 | 5 | 12 | 6 |
| 5 | 12 | 13 | 30 | 30 |
| 8 | 15 | 17 | 40 | 60 |
| 7 | 24 | 25 | 56 | 84 |
| 20 | 21 | 29 | 70 | 210 |
| 12 | 35 | 37 | 84 | 210 |
Practical application of the Pythagorean theorem in construction
The simplest and most frequently used trick in construction based on the Pythagorean theorem is the3-4-5rule, used to determine a right angle in the field without the use of specialized surveying tools:
- A bricklayer measures the distance on one wall 3 meters from the corner and he makes a sign.
- On the other, perpendicular wall, he measures a distance of 4 meters and also makes a mark.
- Then measures the diagonal (straight line) distance between the two characters. If the distance is exactly 5 meters, it means that the corner holds a perfect 90-degree angle. If the distance is different, the angle needs to be adjusted.
Frequently asked questions (FAQ)
What is the formula for the Pythagorean theorem?
The formula is a² + b² = c², where "a" and "b" are the lengths of the legs of a right triangle, and "c" is the length of the hypotenuse.
What are Pythagorean triples?
These are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c² (e.g. 3, 4, 5 or 5, 12, 13).
Does the Pythagorean theorem work in every triangle?
No, this theorem only applies to right triangles (those that have one angle measuring exactly 90 degrees).
How to check if a triangle with sides 6, 8, 10 is a right triangle?
Plug the values into the formula: 6² + 8² = 36 + 64 = 100. Since 10² = 100, the equation is true, so a triangle with these sides is a right triangle.
What are adjoining and hypotenuse?
The legs are the two shorter sides of a triangle that meet to form a right angle. The hypotenuse is the longest side opposite a right angle.