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EXTRA

Unit 53: Area and Volume

On the digital SAT, you will probably see at least one question that asks about area and volume. Fortunately for us, an equation sheet is provided with all the necessary formulas for finding the area and volume of different shapes!

Here is a copy of the reference sheet you will have access to on your exam:

It is important to familiarize yourself with the formulas and reference sheet before your test date so that you won’t waste time during the test figuring out what each one means.

Strategy:

  1. Identify the correct formula to use.
    • Is the problem asking for area? Volume? What geometric shape is the question referring to?
  2. Plug in the correct values
    • Substitute the given values into the formula. Read the question carefully! If it is asking for the radius, make sure to give the radius and not the diameter.

Let’s look at an example problem:

A cylinder has a diameter of 6 inches and a height of 16 inches. What is the volume, in cubic inches, of the cylinder?

A) (96mathrmpi)
B) (144mathrmpi)
C) (576mathrmpi)
D) (2304mathrmpi)

First, we see that we are looking for the volume of a cylinder. Looking at our reference sheet, this means that we will use the equation:

$$V=mathrmpi r^2h$$

Now, we can plug in the correct values into our equation. Notice that our problem equation gives us the diameter of the cylinder, but our equation asks for the radius. Make sure to read the question carefully!! This is an easy place to make mistakes.

We find the radius of our cylinder by dividing the diameter by 2.

(r=frac d2), so (r=frac62=3)

The height of our cylinder is 16 inches.

Plugging this all into our equation:

(V=mathrmpi(3^2)(16))
(V=144mathrmpi)
B is the correct answer.

Let’s look at another example problem that is a little bit trickier.

Square A has side lengths that are 24 times the side lengths of Square B. The area of Square A is m times the area of Square B. What is the value of m?

In this problem we are looking at the area of two different squares, so the formula we use from the reference sheet is (A=lw). Remember that in the case of squares however, both the length and width are the same value so we can rewrite this equation as (A=s^2).

We can compare the area of each square to determine the value of m.

We are told that the side lengths of Square A are 24 times the side length of Square B

We can define the side lengths of Square B as (x)
We can define the side lengths of Square A as (24x)

Using these definitions, let’s compare the areas of both squares:

Square A: (A=s^2 = {(24x)}^2 = 576x^2)
Square B: (A=s^2 = x^2)

We also know that the area of Square A is m times the area of Square B.

(area;A=mcdot area;B)
(576x^2=mcdot x^2)
(m=576)

The value of m is equal to 576.

Perimeter and Surface Area

Once in a while, you may get a question on the digital SAT that deals with perimeter and surface area. These will only apply to squares/rectangles for perimeter, or cubes/prisms for surface area.

The perimeter of an object is the distance around a particular shape. Although it is not on your reference sheet the formula for the perimeter of rectangles is an good one to know:

$$P=2(l+w)$$

The surface area of an object is the total area of the surface of a three-dimensional object. Think of it as the amount wrapping paper needed to cover the entirety of a prism. You can find the surface area by adding up the areas of each side of an object. Again, although it is not on your reference sheet the formula for the surface area of a cube in particular is a good one to know:

$$A=6a^2$$

a = length of the edge