SSAT Upper Level Quantitative - How to find the volume of a prism

Box

The above diagram shows a rectangular solid. The shaded side is a square. Give the volume of the solid.

Explanation

A square has four sides of equal length, so, as seen in the diagram below, the length, width, and height of the solid are 20, 10, and 10.

Box 3

The volume of the solid is the product of the three dimensions:

$V = LWH = 20 \cdot 10 \cdot 10 = 2,000$ .

Box

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the solid.

$30x^{2}$

$2x^{2} + 120$

Explanation

The volume of a rectangular solid is the product of its three dimensions. Since the left (and right) surfaces are squares, as seen in the figure below, the missing dimension is equal to .

Untitled 4

The dimensions are 30, , and , making the volume

$V = 30 \cdot x \cdot x = 30x ^{2}$ .

Find the volume of a circular prism with a base radius of and a height of .

$2420\pi:cm^{3}$

$2420\pi:cm^{2}$

$220\pi:cm^{3}$

$440\pi:cm^{3}$

$440\pi:cm^{2}$

Explanation

The volume of a given prism , where is the base area and is the height. For a circular prism, the base area $B=\pi r^{2}$ , or pi times the square of the radius. Therefore:

$V=(\pi r^{2})h$

Substituting in our known values:

$V=(\pi (11:cm)^{2})(20:cm)$

$V=(121\pi:cm^{2})(20:cm)$

$V=2420\pi:cm^{3}$

A triangular prism has a triangle base length of , a triangle height length of , and a prism height of . What is the volume of the prism?

$192:cm^{3}$

$192:cm^{2}$

$384:cm^{3}$

$384:cm^{2}$

Not enough information provided

Explanation

The volume of a given prism , where is the base area and is the prism height. For a triangular prism, the base area $B=\frac{1}{2}bh_{2}$ , where is the base of the triangle and $h_{2}$ is the height of the triangle (not of the prism).

Therefore, we can substitute the base area equation into the equation for the volume of a prism:

$V=(\frac{1}{2}bh_{2})h$

Substituting in our known values:

$V=(\frac{1}{2}(4:cm)(8:cm))(12:cm)$

$V=(16:cm^{2})(12:cm)$

$V=192:cm^{3}$

A given rectangular prism has a length of , a width of , and a height of , what is its volume?

$1050cm^{3}$

$1050cm^{2}$

$32cm^{3}$

$85cm^{3}$

$525cm^{3}$

Explanation

The volume of a rectangular prism is the product of its base area and its height : . Since we can determine the base area of the rectangular prism from its length and width, we can rewrite the equation and solve:

A given rectangular prism has a length of , a width of , and a height of . What is the volume of the prism?

$315:cm^{3}$

$315:cm^{2}$

$63:cm^{3}$

$63:cm^{2}$

$45:cm^{3}$

Explanation

The volume of a given prism , where is the base area and is the height. For a rectangular prism, the base area $B=l \times w$ , or length times width. Therefore: