How many $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$ blocks are needed to build the solid rectangular prism shown?
(A) 10
(B) 12
(C) 33
(D) 66
(E) 36

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-056.jpg?height=309&width=453&top_left_y=2231&top_left_x=1245)
Solution 1

To build the solid rectangular prism, we could first construct the $4 \mathrm{~cm}$ by $3 \mathrm{~cm}$ base using $4 \times 3=12$ of the $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$ blocks.

Two more layers identical to the first layer, placed on top of the first layer, would give the prism its required $3 \mathrm{~cm}$ height.

This would require 12 more $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$ blocks in layer two and 12 more in layer three, or $12 \times 3=36$ blocks in total.

Solution 2

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-258.jpg?height=417&width=315&top_left_y=1819&top_left_x=1599)

Equivalently, this question is asking for the volume of the rectangular prism.

The volume of a prism is the area of the base times the height,

or $V=4 \mathrm{~cm} \times 3 \mathrm{~cm} \times 3 \mathrm{~cm}=36 \mathrm{~cm}^{3}$.

Since the volume of each of the $1 \mathrm{~cm} \times 1 \mathrm{~cm} \times 1 \mathrm{~cm}$ blocks is $1 \mathrm{~cm}^{3}$, then 36 blocks are needed to build the solid rectangular prism.

(The prism can actually be built with 36 blocks as seen in Solution 1.)

ANSWER: (E)