[fs-toc-h2]Surface Area

If you understand the concept of area, you’ll be fine with this. The surface area of any 3D shape is the sum of the area of each side.

NOTE: So far, there have not been any questions on the surface area of a sphere or cylinder.

Take a look at the example cube below and try to find its surface area.

In order to find the surface area of a cube, you must find the area of one of the square sides first. With a side length of 3 cm, the area of a square side is \(3 \cdot 3\) or simply \(3^2\)

\[
3^2 = 9 \text{ cm}^2
\]

A cube is made up of 6 equal square sides. If the area of one side is 9 cm², then the area of the whole shape would be \(6 \times 9\), as there are 6 sides with an area of 9 cm²

\[6 \times 9 = 54 \text{ cm}^2\]

The surface area of a rectangular prism takes a few more steps to calculate but is still the same concept. Remember, surface area is just the area of each side added up.

In a rectangular prism, there are two equal bases and four sides.

Now in this case, we have to find the area of both the two bases that make up the ends of the prism, and the four rectangular sides. Try to find the surface area of the figure below.

Just like every other unit we’ve gone over, take this one step at a time. Firstly, find the area of the bases (the ends of the prism). In this case the bases have a length of 4 cm and a height of 2 cm, giving us an area of

\[2 \times 4 = 8 \text{ cm}^2\]

Now we’re left with the rectangular sides, we have to deal with these with one set at a time.

These first two rectangles have a length of 12 and a width of 2 cm, making their areas:

\[
A = 12 \times 2
\]

\[
A = 24 \text{ cm}^2
\]

Now we have the area of the second set of rectangles.

These rectangles have a length of 12 and a width of 4 cm, making their areas:

\[
A = 12 \times 4
\]

\[
A = 48 \text{ cm}^2
\]

We now have the area of 4 rectangular sides and 2 bases. Since there are two of each shape, multiply each area by 2, and find the sum in order to get the surface area.

\[SA = (2 \times 8) + (2 \times 24) + (2 \times 48)\]

\[SA = 160 \text{ cm}^2\]

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[fs-toc-h2]Volume

Thankfully, the SAT was kind enough to bless us with an entire formula sheet full of volume formulas. This means that you aren’t expected to memorize any of these, but instead what they mean and how to use them.

The volume of a 3D shape, put simply, is the amount of space within it.

In order to calculate the volume of any figure, simply find the formula on the sheet and input the information given. Try solving the example below.

If a triangular prism has a height of 10 cm, a length of 3 cm, and a width of 7 cm. What is the volume of the prism?

First step is to find the formula, which is given as

\[V = \frac{1}{3} lwh\]

Now plug in each given value to solve for V

\[V = \frac{1}{3} (3)(7)(10)\]

\[V = 70 \text{ cm}^3\]

In some cases you’ll be given a volume, and asked to solve for one of the dimensions.

The process here is the exact same, you’re simply just solving for a different variable. Plug in each value you know and solve for the missing one.

\[V = \frac{1}{3} (\pi) r^2 h\]

\[24(\pi) = \frac{1}{3} (\pi) (r^2)(2)\]

\[24 = \frac{1}{3} (r^2)(2)\]

\[36 = r^2\]

\[6 = r\]

Making choice B the correct answer.

In this case, you simply have to find the volume of each shape, then add all three together. We can find the volume of the two equivalent cones first:

\[V = \frac{1}{3} (\pi) r^2 h\]

\[V = \frac{1}{3} (\pi) (5^2)(5)\]

\[V \approx 130.8 \text{ cm}^3\]

Since both cones are equal, the volume of both cones combined would be \(2 \times 130.8\) or \(262.6 \text{ cm}^3\)

Now find the volume of the cylinder.

\[V = (\pi) r^2 h\]

\[V = (\pi) (5^2)(10)\]

\[V \approx 785 \text{ ft}^3\]

The question wants us to find the full volume of the silo, which is made of these three shapes, so you can add the volumes together.

\[785 \text{ ft}^3 + 262.6 \text{ ft}^3 = 1047.6 \text{ ft}^3\]

This answer is closest to choice D, making that the right answer.

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