If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Area and volume | Lesson

A guide to area and volume on the digital SAT

What are area and volume problems?

Area and volume problems focus on using the relevant formulas for various two- and three-dimensional shapes. We'll be expected to calculate the length, area, surface area, and volume of shapes, as well as describe how changes in side length affect area and volume.
In this lesson, we'll learn to:
  1. Calculate the volumes and dimensions of three-dimensional solids
  2. Determine how dimension changes affect area and volume
You can learn anything. Let's do this!

How do I calculate the volumes and dimensions of shapes?

Volume word problem: gold ring

Khan Academy video wrapper
Volume word problem: gold ringSee video transcript

Volume of a cone

Khan Academy video wrapper
Volume of a coneSee video transcript

The volumes of three-dimensional solids

Good news: You do not need to remember any volume formulas for the SAT! At the beginning of each SAT math section, the following volume formulas are provided as reference.
ShapeFormula
Right rectangular prismV=wh
Right circular cylinderV=πr2h
SphereV=43πr3
Right circular coneV=13πr2h
Rectangular pyramidV=13wh
If the test asks for the volume of a different shape, the volume formula will be provided alongside the question.
To calculate the volume of a solid:
  1. Find the volume formula for the solid.
  2. Plug the dimensions into the formula.
  3. Evaluate the volume.
Example: Fei Fei has a model of the Moon in the shape of a sphere. If the model has a radius of 10 centimeters, what is the volume of the model in cubic centimeters?
Some questions will provide the volume of the solid and ask us to find a linear dimension such as length or radius.
To find an unknown dimension when the volume of a solid is given:
  1. Find the volume formula for the solid.
  2. Plug the volume and any known dimensions into the formula.
  3. Isolate the unknown dimension.
Example: A puzzle box is shaped like a rectangular prism and has a volume of 240 cubic inches. If the puzzle box has a length of 10 inches and a width of 8 inches, what is the height of the puzzle box in inches?

Try it!

try: find the volume of a pyramid
A pyramid has a square base with a side length of 8 centimeters. The height of the pyramid is 34 as long as the side length of its base.
What is the height of the pyramid in centimeters?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the volume of the pyramid in cubic centimeters?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

How do changing dimensions affect area and volume?

How volume changes when dimensions change

Khan Academy video wrapper
How volume changes from changing dimensionsSee video transcript

Impact of increasing the radius

Khan Academy video wrapper
Impact of increasing the radiusSee video transcript

The effect of changing dimensions on area and volume

When a linear dimension to the first power, e.g., the length of a rectangle or the height of a cylinder, changes by a factor, the area or volume changes by the same factor.
However, when a linear dimension to the second power, e.g., the side length of a square or the radius of a cylinder or cone, changes by a factor, the area or volume changes by the square of the factor.

Try it!

try: compare the volumes of two cylinders
Right circular cylinder A has a volume of 64π cubic feet. Which of the following right circular cylinders have the same volume as cylinder A ?
Choose 2 answers:

Your turn!

practice: calculate a volume
What is the volume, in cubic meters, of a right rectangular prism that has a length of 2 meters, a width of 0.4 meter, and a height of 5 meters?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

practice: calculate a linear dimension
A medicine bottle is in the shape of a right circular cylinder. If the volume of the bottle is 144π cubic centimeters, what is the diameter of the base of the bottle, in centimeters?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

practice: determine the effect of scaling on volume
The volume of a right circular cone A is 225 cubic inches. What is the volume, in cubic inches, of a right circular cone with twice the radius and twice the height of cone A?
Choose 1 answer:

Want to join the conversation?