Pythagoras, Pythagorean Theorem, 3D Models

The Pythagorean Theorem can be illustrated (not proved) using 3D models. In fact, it is a direct implication of a^2 + b^2 = c^2 in a 3D space. Pick an h > 0. Then, ha^2 + hb^2 = hc^2. We could construct a square-end box of a constant height along each side of the right triangle. However, there is much more to it. We could build infinitely many 3D (or 2D) models. As long as the three 3D solids (or corresponding 2D shapes) are similar to each other, the volume of the solid on the hypotenuse is equal to the sum of the volumes of the two solids on the two legs. Some 3D models are aesthetically attractive; others are not so much.

Included here are three 3D models on a right triangle that is 30mm, 50mm, and sqrt(3400)mm—semi-circle, square, and trapezoid. They all have a depth of 30mm. You can use water, sand, or rice to demonstrate the relationships. It is a bit messy and is also fun for students!

No

No

.1 to .2mm

10-25%

By default, the models are 30mm in depth. Please feel free to scale the height down. It will not affect its properties.