Question 1

What is the Pythagorean theorem?

Accepted Answer

In a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c². The hypotenuse is always the longest side and always opposite the right angle. Rearranged, a missing leg is √(c² − a²) — which is why the calculator solves in both directions.

Question 2

Does it work for triangles without a right angle?

Accepted Answer

No, and this is the most common misapplication. For any other triangle you need the law of cosines, c² = a² + b² − 2ab·cos(C), which reduces to Pythagoras when C is 90° because cos 90° is zero. If your triangle has no right angle, using a² + b² = c² will give a confidently wrong answer.

Question 3

What is a Pythagorean triple?

Accepted Answer

Three whole numbers satisfying the theorem — 3-4-5, 5-12-13, 8-15-17 and their multiples. Builders use the 3-4-5 method to square a corner without a set square: measure 3 units along one wall, 4 along the other, and adjust until the diagonal is exactly 5. It is one of the oldest practical applications of mathematics still in daily use.

Question 4

How do I check whether a corner is truly square?

Accepted Answer

Use the 3-4-5 method, scaled to whatever units suit the job — 30 cm, 40 cm and 50 cm for a small frame, or 3 m, 4 m and 5 m for a foundation. If the diagonal measures more than 5, the angle is obtuse; less, and it is acute. Larger triangles give better precision because the same measuring error represents a smaller angular deviation.

Question 5

Does it extend to three dimensions?

Accepted Answer

Yes. The space diagonal of a box is √(l² + w² + h²) — Pythagoras applied twice, once across the base and once up to the far corner. This is how you work out whether a long object fits inside a container diagonally, and it generalises to any number of dimensions, which is why it underpins distance calculations in data science.

Question 6

Why can the hypotenuse not be shorter than a leg?

Accepted Answer

Because c² = a² + b², and adding a positive number to a² always gives something larger. If your calculation produces a hypotenuse shorter than one of the legs, you have either mislabelled which side is the hypotenuse or made an arithmetic error. It is a useful automatic check on your working.

Hypotenuse Calculator

Everyday Uses

Finding a missing side

Squaring a corner

Diagonal fit

Straight-line distance

Screen diagonals

Ladder safety

Frequently Asked Questions

What is the Pythagorean theorem?

Does it work for triangles without a right angle?

What is a Pythagorean triple?

How do I check whether a corner is truly square?

Does it extend to three dimensions?

Why can the hypotenuse not be shorter than a leg?

Related Calculators

Everyday Uses

Finding a missing side

Squaring a corner

Diagonal fit

Straight-line distance

Screen diagonals

Ladder safety

Frequently Asked Questions

What is the Pythagorean theorem?

Does it work for triangles without a right angle?

What is a Pythagorean triple?

How do I check whether a corner is truly square?

Does it extend to three dimensions?

Why can the hypotenuse not be shorter than a leg?