The Pythagorean theorem: a² + b² = c²
Few formulas are as famous and as useful as the Pythagorean theorem. It holds in every right triangle and links its three sides: the sum of the squares of the two legs equals the square of the hypotenuse.
The legs a and b are the two sides that form the right angle. The hypotenuse c lies opposite the right angle and is always the longest side. In the volume calculator, the "Pythagoras" option is a triangle solver that works out all sides and angles from any two known values.
Example 1: finding the hypotenuse
Given a = 3 cm and b = 4 cm: c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm. This 3-4-5 triangle is the best-known example because all sides are whole numbers. Other such triples are 5-12-13, 8-15-17 and all multiples, such as 6-8-10.
Example 2: finding a leg
A 5 m ladder leans against a house wall. Its foot stands 1.5 m from the wall. How high does it reach? The ladder is the hypotenuse, the distance is one leg. Rearranged: a = √(c² − b²) = √(25 − 2.25) = √22.75 ≈ 4.77 m. When rearranging, the plus becomes a minus — and the hypotenuse must always be longer than the leg, otherwise there is no solution.
Example 3: screen or room diagonal
A screen diagonal is the hypotenuse of width and height. A monitor 60 cm wide and 34 cm high has a diagonal of √(60² + 34²) = √4,756 ≈ 69 cm, about 27 inches. The same principle works in three dimensions: the space diagonal of a cuboid is d = √(l² + w² + h²). That tells you whether a long pole will still fit into a van.
Example 4: setting out a right angle
Builders use the 3-4-5 rule to mark an exact right angle on site: measure 3 m along one side and 4 m along the other — if the diagonal is exactly 5 m, the corner is square. For larger areas use multiples such as 6-8-10 m.
Finding angles with sine, cosine and tangent
Pythagoras gives you the sides, trigonometry the angles. For the angle α opposite leg a:
1. sin(α) = a ÷ c
2. cos(α) = b ÷ c
3. tan(α) = a ÷ b
For the 3-4-5 triangle, tan(α) = 3 ÷ 4 = 0.75, so α ≈ 36.87°. The second acute angle follows from the angle sum: β = 90° − 36.87° = 53.13°. Conversely, from one side and one angle you can find the other sides, for example a = c · sin(α).
Pythagoras in solids
The theorem keeps appearing in volume and surface calculations: the slant height of a cone is s = √(r² + h²), the slant height of a pyramid face hₛ = √(h² + (a/2)²). The guide How to calculate volume: every formula gives an overview of all solid formulas.
Conclusion
The Pythagorean theorem only applies to right triangles, but there it always applies. Remember: the hypotenuse stands alone on one side of the equation, and it is the longest side. Sine, cosine and tangent add the angles. To check your work, two known values in the triangle solver are enough.