All ISEE Upper Level Math Resources
Example Questions
Example Question #1 : Solid Geometry
A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.
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 #1 : 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?
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 #4 : Volume
The height of a right pyramid is feet. Its base is a square with sidelength feet. Give its volume in cubic inches.
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 #1 : 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.
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 : Volume
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.
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?
The volume of a pyramid can be determined using the following equation:
Example Question #2 : How To Find The Volume Of A Pyramid
A right regular pyramid with volume has its vertices at the points
where .
Evaluate .
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 #3 : Pyramids
Find the volume of a pyramid with the following measurements:
- length = 4in
- width = 3in
- height = 5in
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
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 : Pyramids
Find the volume of a pyramid with the following measurements:
- length: 7in
- width: 6in
- height: 8in
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