Pythagorean theorem explorer

What the tool shows

In a right triangle, the two shorter sides that make the right angle are the legs, a and b, and the longest side, opposite the right angle, is the hypotenuse, c. The Pythagorean theorem says a² + b² = c². On this page, drag the coral and blue handles, or use the steppers, to set each leg from 1 to 12 units. A square is drawn on every side, and each square is labeled with its area.

Unit squares draws a grid inside the squares on the legs, and inside the tilted square on the hypotenuse whenever c is a whole number, so the areas can be counted. The readout adds the two areas and finds c. Hide c turns that part off. The triple finder keeps a list of every set of whole-number sides you find.

Why the squares matter

The squares turn the theorem into a statement about area: the area of the square on the hypotenuse equals the areas of the squares on the two legs added together. That is why the sides are squared. Common Core standard 8.G.B.6 asks students to explain a proof of the theorem and its converse, and 8.G.B.7 asks them to use it to find unknown side lengths. Counting squares for a 3, 4, 5 triangle is a concrete first step before algebra, and seeing that most hypotenuses are not whole numbers explains why square roots come up.

Common mix-ups

• Adding the legs. A triangle with legs 3 and 4 has a hypotenuse of 5, not 7; it is the squares that add.
• Forgetting the square root. a² + b² gives c², not c. For legs 3 and 4, c² is 25 and c is 5.
• Using the theorem on any triangle. It only works when there is a right angle.
• Putting the hypotenuse in the wrong place. c is always the longest side, opposite the right angle.

A 10-minute lesson

• Set the legs to 3 and 4 and count the squares on each side together.
• Turn on Hide c. Give legs of 5 and 12 and ask for c before checking.
• Try legs of 2 and 3. Ask why the hypotenuse is between 3 and 4 and what √13 means.
• Use the triple finder to look for more triples. Hint: try doubling a triple you found.
• Finish by asking the student to state the theorem in words, using the word area.

Where it fits by grade

• Grade 8: explain a proof of the theorem and its converse (8.G.B.6), find unknown sides in two and three dimensions (8.G.B.7), and find the distance between two points (8.G.B.8).
• High school: the theorem underpins distance, trigonometry and the equation of a circle.
• England: Pythagoras' theorem is taught in Key Stage 3, often in Year 8 or 9, and used throughout GCSE Maths.

Learning more

Exponents and square roots are on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Exam Prep includes GCSE Maths practice papers, and when a Pythagoras problem is tricky the AI tutor asks guiding questions so the student does the reasoning.