Pythagorean Theorem Worksheets

Every question comes with a diagram

Each problem prints a small right triangle with the known sides labelled and the unknown marked, which is the difference between a worksheet and a list of numbers.

Twelve problems by default, sides up to 20, and an answer key.

Two directions

Find the hypotenuse is the forward case: square both legs, add, take the root. Straightforward once the formula is known.

Find a leg is the one that catches people. Given c and one leg, you subtract rather than add — a² = c² − b² — and pupils who have only met the forward case will confidently add anyway and produce an answer longer than the hypotenuse. The sketch is what makes that error visible; a number longer than the longest side is obviously wrong once you can see the triangle.

Mixed mode alternates, which is where the checking habit gets built.

Triples, and why they are favoured

Most problems are drawn from Pythagorean triples — 3-4-5, 5-12-13, 8-15-17, and their multiples — so the answers come out whole.

Some find-the-hypotenuse questions use arbitrary legs instead, and those answers round to one decimal place. That mixture is deliberate: a pupil who only ever meets triples starts to believe every triangle has an integer hypotenuse, which is spectacularly untrue and worth breaking early.

Recognising 3-4-5 and its family on sight is worth having anyway. It is the fastest sanity check available on any right-triangle question.

The mistakes to expect

Forgetting the square root at the end, which produces an answer that is the square of the truth. Adding when the missing side is a leg. And identifying the wrong side as the hypotenuse, which is always the one opposite the right angle regardless of how the triangle is drawn or rotated.

That last one is why the diagrams are not all drawn the same way up.

Where it goes next

Distance between two points on a grid is Pythagoras with the legs read off the axes. So is the diagonal of a rectangle, the height of a ladder against a wall, and the shortest route across a field. Trigonometry is the same triangle with angles added.

It is one of the few pieces of school mathematics that people genuinely use afterwards — builders check a corner is square with a 3-4-5 measurement, and have done for several thousand years.

Level

Typically ages thirteen to sixteen. The find-the-leg mode and the non-integer answers push it towards the upper end of that.

Related: squares and roots, which is the fluency this depends on.