How to use the Pythagorean Theorem
- Enter the two side lengths you know.
- Leave the unknown side blank.
- Read the missing side plus area, perimeter and the two acute angles.
- Check that the hypotenuse is the longest side — if not, the inputs are mislabelled.
How the calculation works
a² + b² = c², with c always the side opposite the right angle. To find a leg, rearrange to a = √(c² − b²); if that expression is negative your supposed hypotenuse is shorter than a leg, which is geometrically impossible. Angles follow from basic trigonometry: θ = arctan(opposite ÷ adjacent).
The theorem generalises directly into the distance formula: the gap between two points is √((x₂−x₁)² + (y₂−y₁)²), which is just Pythagoras applied to the horizontal and vertical differences. It extends to three dimensions by adding a z term, which is how diagonal spans of boxes and rooms are found. Builders use the 3-4-5 special case in reverse to square up corners on site.
c = √(a² + b²) ; a = √(c² − b²)Source: Euclid's Elements, Book I, Proposition 47 — the Pythagorean theorem.
Worked example
A ladder reaching 5.6 m up a wall with its base 2.1 m out from it.
- a = 5.6, b = 2.1.
- a² + b² = 31.36 + 4.41 = 35.77.
- c = √35.77 = 5.98 m.
- Angle at base = arctan(5.6 / 2.1) = 69.4°.
The ladder is about 5.98 m long, leaning at 69.4° — steeper than the 75° maximum many safety guides allow.
Frequently asked questions
Does it work for non-right triangles?+
No. Use the law of cosines, which reduces to Pythagoras when the angle is exactly 90°.
What is a Pythagorean triple?+
A set of three integers satisfying the theorem, such as 3-4-5, 5-12-13 and 8-15-17.
How do I check a corner is square?+
Measure 3 units along one edge and 4 along the other; if the diagonal is exactly 5, the corner is square.
How does this extend to 3D?+
The space diagonal of a box is √(l² + w² + h²) — the theorem applied twice.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.