Question

Given the cube below, which equation can be used to prove the volume of a pyramid through dissection of the cube?

c x 7 x 3 1/2 x 3 1/2 = 1/6(343)

c x 7 x 7 x 3 1/2 = 1/6(343)

c x 3 1/2 x 3 1/2 x 3 1/2 = 343

c x 7 x 7 x 7 = 343

Answer 1

c × 7 × 7 × 3½ = 1/6(343)

Explanation:

Draw the four diagonals of the cube.

They divide the cube into six identical pyramids. I have outlined one of them for you.

The volume of the cube is

V = 7 cm × 7 cm × 7 cm = 643 cm³

There are six pyramids, so the volume of one pyramid is

Vₚ = ⅙ × 7 cm × 7 cm × 7 cm = ⅙(343) cm³

Also, the volume of a pyramid is directly proportional to the area of the base and to the height.

Vₚ = c × A × h, where c is some constant

A = 7 cm × 7 cm, and h = 3½ cm. So,

Vₚ = c × 7 cm × 7 cm × 3½ cm = ⅙(343) cm³