### All ISEE Upper Level Math Resources

## Example Questions

### 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.

**Possible Answers:**

**Correct answer:**

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 : Solid Geometry

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

**Possible Answers:**

**Correct answer:**

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 #1 : Volume

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

**Possible Answers:**

**Correct answer:**

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 #3 : Solid Geometry

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

**Possible Answers:**

**Correct answer:**

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 #1 : Solid Geometry

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.

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find The Volume Of A Pyramid

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

**Possible Answers:**

**Correct answer:**

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

### Example Question #1 : Pyramids

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

where .

Evaluate .

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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 #1 : Pyramids

Find the volume of a pyramid with the following measurements:

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

**Possible Answers:**

**Correct answer:**

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 #2 : How To Find The Volume Of A Pyramid

Find the volume of a pyramid with the following measurements:

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

**Possible Answers:**

**Correct answer:**

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: 7in
- width: 6in
- height: 8in

So, we get