Move all pieces from squares A and B into square C. Tap a piece to rotate it.
Area A²
9
+
Area B²
16
=
Area C²
25
The Geometry of a Theorem
The Pythagorean Theorem states that in any right-angled triangle, the area of the square on the hypotenuse (the longest side, C) is exactly equal to the sum of the areas of the squares on the other two sides (A and B). This is famously written algebraically as a² + b² = c².
Inquiry Questions:
Count the background grid squares inside the pink and green areas. Do they perfectly match your calculated areas?
What happens to the total area (C²) when you increase Side A but decrease Side B?
Can you find a combination on the sliders where C² is a perfect square (an integer multiplied by itself, like 9, 16, or 25)?
✨
Proof Complete!
a² + b² = c²
Drag pieces to the dashed square.
Perigal's Epitaph
Henry Perigal (1801–1898) was an English stockbroker and amateur mathematician. He is famous for discovering this elegant "dissection proof" in 1891. He was so incredibly proud of this purely visual, slide-and-rotate geometric proof that he requested it be carved onto his tombstone in Essex as his epitaph.
Inquiry Questions:
Notice how the green square (B²) is cut. The cuts pass directly through its center, parallel and perpendicular to the hypotenuse. Why are these specific angles necessary?
How does this physical puzzle prove the theorem without using any numbers, algebra, or measurements?
🏆
Theorem Proven!
a² + b² exactly tiles c².
Move all pieces into square C. Tap to rotate.
Pythagorean Triples
While the Pythagorean theorem works for all right triangles, it is especially satisfying when all three side lengths are perfect whole numbers. These rare sets of integers—like 3, 4, and 5 or 5, 12, and 13—are mathematically known as Pythagorean Triples. Because they are whole numbers, we can perfectly tile their areas using uniform 1x1 grid blocks.
Inquiry Questions:
Count the individual 1x1 unit squares inside the colored blocks. How does counting them physically connect to the algebraic equation a² + b² = c²?
If you had a right triangle with sides 6 and 8, how many 1x1 square units would be required to perfectly tile the largest square? (Is 6-8-10 a Pythagorean Triple?)