ISEE Upper Level Quantitative Reasoning - How to find the volume of a pyramid

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Explanation

Use the formula $V = \frac {1}{3} Bh$ on each pyramid.

(a) $B = x^{2}; h = 2x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot x^{2} \cdot 2x = \frac {2}{3} x^{3}$

(b) $B = (2 x)^{2} = 4x^{2}; h = x$

$V = \frac {1}{3} Bh = \frac {1}{3} \cdot 4x^{2} \cdot x = \frac {4}{3} x^{3}$

Regardless of , (b) is the greater quantity.

Find the volume of a pyramid with the following measurements:

length: 7in
width: 6in
height: 8in

$112\text{in}^3$

$336\text{in}^3$

$221\text{in}^3$

$168\text{in}^3$

$426\text{in}^3$

Explanation

To find the volume of a pyramid, we will use the following formula:

$V = \frac{lwh}{3}$

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

length: 7in
width: 6in
height: 8in

So, we get

$V = \frac{ 7\text{in} \cdot 6\text{in} \cdot 8\text{in}}{3}$

$V = \frac{42\text{in}^2 \cdot 8\text{in}}{3}$

$V = \frac{336\text{in}^3}{3}$

$V = 112\text{in}^3$

Find the volume of a pyramid with the following measurements:

length = 4in
width = 3in
height = 5in

$20\text{in}^3$

$60\text{in}^3$

$40\text{in}^3$

$80\text{in}^3$

$12\text{in}^3$

Explanation

To find the volume of a pyramid, we will use the following formula:

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

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the base of the pyramid has a length of 4in. We also know the base of the pyramid has a width of 3in. We also know the pyramid has a height of 5in.

Knowing this, we can substitute into the formula. We get

$V = \frac{4\text{in} \cdot 3\text{in} \cdot 5\text{in}}{3}$

$V = \frac{60\text{in}^3}{3}$

$V = 20\text{in}^3$

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.

Find the volume of a pyramid with the following measurements:

length = 4cm
width = 9cm
height = 8cm

$96\text{cm}^3$

$288\text{cm}^3$

$24\text{cm}^3$

$126\text{cm}^3$

$154\text{cm}^3$

Explanation

To find the volume of a pyramid, we will use the following formula:

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

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the following measurements:

length = 4cm
width = 9cm
height = 8cm

Knowing this, we can substitute into the formula. We get

$V = \frac{4\text{cm} \cdot 9\text{cm} \cdot 8\text{cm}}{3}$

$V = \frac{288\text{cm}^3}{3}$

$V = 96\text{cm}^3$

A foot tall pyramid has a square base measuring feet on each side. What is the volume of the pyramid?

$160,000 ft^{3}$

$4,000 ft^{3}$

$160,000 ft^{2}$

$480,000 ft^{3}$

$12,000 ft^{3}$

Explanation

In order to find the area of a triangle, we use the formula $\frac{1}{3}Bh$ . In this case, since the base is a square, we can replace with $s^{2}$ , so our formula for volume is $\frac{1}{3}s^{2}h$ .