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### Course: Digital SAT Math>Unit 13

Lesson 1: Area and volume: advanced

# 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

Volume word problem: gold ringSee video transcript

### Volume of a cone

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 prism$V=\ell wh$
Right circular cylinder$V=\pi {r}^{2}h$
Sphere$V=\frac{4}{3}\pi {r}^{3}$
Right circular cone$V=\frac{1}{3}\pi {r}^{2}h$
Rectangular pyramid$V=\frac{1}{3}\ell wh$
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 $\frac{3}{4}$ as long as the side length of its base.
What is the height of the pyramid in centimeters?
What is the volume of the pyramid in cubic centimeters?

## How do changing dimensions affect area and volume?

### How volume changes when dimensions change

How volume changes from changing dimensionsSee video transcript

### Impact of increasing the radius

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 $64\pi$ cubic feet. Which of the following right circular cylinders have the same volume as cylinder $A$ ?

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?

practice: calculate a linear dimension
A medicine bottle is in the shape of a right circular cylinder. If the volume of the bottle is $144\pi$ cubic centimeters, what is the diameter of the base of the bottle, in centimeters?

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 $A$?

## Want to join the conversation?

• Calculating areas and volumes are so much easier when they give you the formula. Schools in my country make students remember the formula
• In my country you have to learn it except during national exams where they give it to you.
• if there was sat art, sal would get 800
• What is the best way to remember the formulas to each shape?
• The best thing is you don't have to remember! All the formulas will be given in the SAT question paper.
• Perfect I have the formulas, no need to think!!
• who is preparing for december SAT? any tips for english section? Is Erica Meltzer's Guide worth reading?
• I'm taking it in October (1 week) and fr I've 0 tips. I will pray whoever up there to get over 1490
• can someone explain the last question pls
• The last question should be more explained.