Pythagoras’ Theorem: An Interactive Demonstration
This interactive demonstration explores Pythagoras’ theorem, the relationship between the three sides of a right-angled triangle: a² + b² = c². The square on the hypotenuse (c) has the same area as the two squares on the shorter sides (a and b) added together. 📐
Grab and drag the blue and green vertices to change the two legs of the triangle, then watch all three squares resize with it. The legs are constrained between lengths of 1 and 5, and the right angle stays fixed.
How to Use
Tab Navigation
Switch between the proof, the tiling, a calculation challenge, and styling:
- Dissection Proof ▭: Watch the squares rearrange to prove the theorem.
- Pythagorean Tiling ▦: See why the theorem holds across the whole plane.
- Problem 🧮: Find a missing side step by step.
- Style 🎨: Change the background and the proof-piece colours.
▭ Dissection Proof
The Dissection Proof tab gives you a hands-on, translation-only (Perigal-style) proof: the square on the larger leg is cut into four pieces that slide — without rotating — to fill the space around the smaller leg’s square, exactly filling the square on the hypotenuse.
🧭 How to Use the Tool
Start with the plain three-square diagram and drag a vertex to change the triangle — notice how all three squares resize together. Toggle Show Cuts to reveal the two lines that divide the larger square into its four pieces.
Now turn on Show Dissection and slowly drag the Dissection slider to scrub the animation. Watch the four pieces slide across — together with the smaller square they fill the hypotenuse square with no gaps and no overlaps. What does that tell you? The two smaller squares have exactly the same area as the largest, so a² + b² = c².
Toggle the Ruler if you want to measure the sides on the canvas for yourself.
▦ Pythagorean Tiling
The Pythagorean Tiling tab tiles the plane with squares of side a and side b, showing that the relationship isn’t a one-off — it holds everywhere.
🧭 How to Use the Tool
Turn on the c-grid overlay to lay a square grid of side c over the tiling. Look at how a single c-square always covers exactly one a-square plus one b-square’s worth of area.
Then use Highlight bridge cell to dim the rest and focus on one c-tile. The pieces inside it are the same five you met in the dissection proof — the tiling and the dissection are two views of the very same idea.
🧮 Problem
The Problem tab hides the squares and labels the triangle a, b, and c so you can practise the calculation.
🧭 How to Use the Tool
Click Randomize for a fresh triangle, and use the problem dropdown to choose whether you are finding a, b, c, or a random side. The two known sides and the formula are given; the unknown side is hidden for you to work out.
Try to find the missing side yourself first, then check your method step by step with the reveals: the side lengths a, b and c, the formula, and the working. Revealing them in order is a good way to spot exactly where a calculation went wrong.
🎨 Style
Use the Style tab to customise the canvas and the proof pieces.
- Paper Style: The Paper Style dropdown switches the background between plain, grid lines, or dots.
- Fill Opacity: The Fill Opacity slider sets how solid the dissection and tiling fills appear.
- Piece Colours: Give each of the four dissection pieces its own colour with the Piece 1, Piece 2, Piece 3, and Piece 4 pickers.
Related Activities
Pythagoras’ theorem connects closely with other geometry and area topics. Explore these related demonstrations: