Question

Sam glued 21 identical cubes together to form the solid shown below. He painted the exposed faces of the solid including the base. Find the total area that Sam painted. ​

Answer 1

`3456\; \rm cm^(2)`.

Explanation:

The diagram suggests that the length of a row of five cubes is
`40\; \rm cm`. Hence, the length of the edge of each cube would be:

`\displaystyle (40\; \rm cm )/(5) = 8\; \rm cm`.

That would correspond to an area of
`(8\; \rm cm)^(2) = 64\; \rm cm^(2)` for each face of the cube.

Refer to the diagram attached.

Viewing from the front and from the rear each shows
`11` faces of the cubes.

Viewing from the top and from the bottom each shows
`10` faces of the cubes.

Viewing from the left and from the right each shows
`6` faces of the cubes.

When viewed from the six perspectives,
`2 * 11 + 2 * 10 + 2 * 6 = 54` faces of the cubes are visible in total.

For this particular construction, all faces that need to be painted are visible when viewed from exactly one of the six perspectives: front, rear, top, right, left, right. Hence, Sam would need to paint
`54` of such
`64\; \rm cm^(2)` squares.

The area that Sam needs to paint would be
`54 * (64\; \rm cm^(2)) = 3456\; \rm cm^(2)`.