# ISEE Upper Level Math : How to find the volume of a pyramid

## Example Questions

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### Example Question #1 : How To Find The Volume Of A Pyramid

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Explanation:

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet =  inches

Sidelength of the base: 3 feet =  inches

The volume of a pyramid is

Since the base is a square, we can replace :

Substitute

The pyramid has volume 20,736 cubic inches.

### Example Question #2 : How To Find The Volume Of A Pyramid

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

Explanation:

In order to find the area of a triangle, we use the formula .  In this case, since the base is a square, we can replace with , so our formula for volume is .

Since the length of each side of the base is feet, we can substitute it in for .

We also know that the height is feet, so we can substitute that in for .

This gives us an answer of .

It is important to remember that volume is expressed in units cubed.

### Example Question #3 : How To Find The Volume Of A Pyramid

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

Explanation:

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

Height:  inches

Sidelength of base:  inches

The base of the pyramid has area

square inches.

Substitute   into the volume formula:

cubic inches

### Example Question #4 : How To Find The Volume Of A Pyramid

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

Explanation:

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

Height:  feet

Sidelength:  feet

The base of the pyramid has area

square feet.

Substitute   into the volume formula:

cubic feet

### Example Question #5 : How To Find The Volume Of A Pyramid

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.

Explanation:

The perimeter of the square base,  feet, is equivalent to  inches; divide by  to get the sidelength of the base - and the height:  inches.

The area of the base is therefore  square inches.

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

cubic inches.

### Example Question #6 : How To Find The Volume Of A Pyramid

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

Explanation:

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

### Example Question #7 : How To Find The Volume Of A Pyramid

A right regular pyramid with volume  has its vertices at the points

where .

Evaluate

Explanation:

The pyramid has a square base that is  units by  units, and its height is  units, as can be seen from this diagram,

The square base has area ; the pyramid has volume

Since the volume is 1,000, we can set this equal to 1,000 and solve for :

### Example Question #8 : How To Find The Volume Of A Pyramid

Find the volume of a pyramid with the following measurements:

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

Explanation:

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

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

### Example Question #9 : How To Find The Volume Of A Pyramid

Find the volume of a pyramid with the following measurements:

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

Explanation:

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

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

### Example Question #10 : How To Find The Volume Of A Pyramid

Find the volume of a pyramid with the following measurements:

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

Explanation:

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

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

Now, we know the following measurements:

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

So, we get

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