Pythagoras Proved & The Volume Pour

Watch the most famous theorem in mathematics prove itself with squares you can resize — then put it to work on ladders, maps, and shipping boxes. And when the shapes go 3D, pour cones into cylinders and discover where every volume formula comes from.

Grade 8 · Unit 6 8.G.6 Proof & converse 8.G.7 Unknown side lengths 8.G.8 Distance in the plane 8.G.9 Cylinder, cone & sphere volume

Part 1

The Living Proof

A square grows on every side of a right triangle. Slide the legs and watch the areas: the two leg-squares always add up to exactly the hypotenuse-square. That's not a coincidence — that's the theorem, alive.

Leg a 3

Leg b 4

Part 2

Right-Angle Casework

Five field cases for the theorem. Lock in your prediction first — then the reveal runs the squares.

Case 1 of 5

Part 3

The Workshop Machines

Three machines: solve for any side, watch two full proofs unfold, and pour your way to every volume formula in the unit.

Hypotenuse, leg, distance, or converse — four jobs, one theorem.

Pick a job:

8.G.6 asks you to EXPLAIN a proof — so here are two, plus the converse, one step at a time.

Pick a proof:

Every 3D formula here comes from a pour. Fill, compare, and the formulas stop being magic.

Pick a pour:

Part 4

Master Builder's Quiz

Ten questions from flat triangles to filled spheres. Wrong answers get a hint naming the exact machine or pour that proves the right idea.

📦 Word Box

Right angle: A 90° angle — the corner the whole theorem is built around.
Hypotenuse (c): The side opposite the right angle — always the longest side of a right triangle.
Legs (a, b): The two sides that form the right angle.
Pythagorean Theorem: In a right triangle, a² + b² = c²: the leg-squares together exactly fill the hypotenuse-square.
Converse: The theorem run backward: if a² + b² = c² for a triangle's sides, the angle between a and b must be right.
Pythagorean triple: Three whole numbers satisfying a² + b² = c², like 3-4-5, 5-12-13, and 20-21-29.
Distance formula: The theorem in disguise: the distance between two points is the hypotenuse over legs Δx and Δy.
Space diagonal: The longest rod that fits in a box — found by using the theorem twice (or √(l² + w² + h²) once).
Cylinder: A circular prism. Volume V = πr²h: the base area, stacked h high.
Cone: A circle tapering to a point. Volume V = ⅓πr²h — exactly one-third of its matching cylinder.
Sphere: All points at distance r from a center. Volume V = ⁴⁄₃πr³.
Radius (r): Center to edge. Careful: formulas square or cube the radius, never the diameter.
Volume: The space inside a 3D shape, measured in cubic units — what the pour measures.