Pre-Algebra - Volume of a Pyramid

Find the volume of a pyramid with a length of 2, width of 6, and a height of 9.

Explanation

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (\textup{ Length} \times\textup{ Width} \times\textup{ Height})$

Substitute the given length, width, and height.

$V=\frac{1}{3} (2\times6\times9)$

Rewrite the inside the parentheses as a factor of .

$V=\frac{1}{3} (2\times (3 \times 2)\times9)$

Cancel the fraction with the three and multiply the terms to get the volume.

$V=2\times 2\times 9 = 36$

The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.

$10,368 \textrm{ in}^{3}$

$31,104 \textrm{ in}^{3}$

$15,552 \textrm{ in}^{3}$

$62,208\textrm{ in}^{3}$

$41,472 \textrm{ in}^{3}$

Explanation

Convert each of the measurements from feet to inches by multiplying by .

Height: $2 \times 12 = 24$ inches

Sidelength of base: $3 \times 12 = 36$ inches

The base of the pyramid has area

$B = s^{2} = 36^{2} = 1,296$ square inches.

Substitute into the volume formula:

$V = \frac {1}{3} Bh$

$V = \frac {1}{3} \cdot 1,296 \cdot 24 =10,368$ cubic inches

What is the volume of a pyramid with the following measurements?

$length=7; width=6;height=9;slant\ length=11$

Explanation

The volume of a pyramid can be determined using the following equation:

$V=\frac{1}{3}lwh=\frac{1}{3}(7)(6)(9)=126$

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

$243 \textrm{ in}^{3}$

$729 \textrm{ in}^{3}$

$576 \textrm{ in}^{3}$

$27 \textrm{ in}^{3}$

$81 \textrm{ in}^{3}$

Explanation

The perimeter of the square base, feet, is equivalent to $3 \times 12 = 36$ inches; divide by to get the sidelength of the base - and the height: $36 \div 4 = 9$ inches.

The area of the base is therefore $B = s^{2} = 9^{2} = 81$ square inches.

In the formula for the volume of a pyramid, substitute :

$V = \frac{1}{3} Bh = \frac{1}{3}\cdot 81 \cdot 9 = 243$ cubic inches.

The pyramid has a length, width, and height of respectively. What is the volume of the pyramid?

Explanation

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (L\times W\times H)$

Substitute the dimensions and solve.

$V=\frac{1}{3} (2\times 4\times 6) = \frac{1}{3}(48)=16$

Find the volume of a pyramid if the length, base, and height are $\textup{10, 20, and 40,}$ respectively.

$\frac{8000}{3}$

$\frac{800}{3}$

$\frac{400}{3}$

Explanation

Write the formula for the volume of a pyramid.

$V=\frac{1}{3} (LWH)$

Substitute the dimensions and solve for the volume.

$V=\frac{1}{3} (10)(20)(40) = \frac{8000}{3}$

Find the volume of a pyramid with a length of 4, width of 7, and a height of 3.

Explanation

Write the formula to find the area of a pyramid.

$V=\frac{1}{3} (L\times W \times H)$

Substitute the dimensions.

$V=\frac{1}{3} (4\times 7\times 3) = 28$

Find the volume of a pyramid with a length of 6cm, a width that is half the length, and a height that is two times the length.

$72\text{cm}^3$

$216\text{cm}^2$

$108\text{cm}^3$

$108\text{cm}^2$

$36\text{cm}^3$

Explanation

The formula for volume of a pyramid is

$V =\frac{ l \cdot w \cdot h}{3}$

where l is the length, w is the width, and h is the height. We know the length is 6cm. The width is half the length, so the width is 3cm. The height is two times the length, so the height is 12cm. Using this information, we substitute. We get

$V =\frac{ 6\text{cm} \cdot 3\text{cm} \cdot 12\text{cm}}{3}$