Solids refer to three-dimensional shapes. On the SAT, the only solids you will have to deal with are cubes, rectangular solids, and right cylinders. For these shapes, you will have to know how to calculate surface area and volume.

A rectangular solid is a six-sided shape in which all angles are 90º. It has length, depth, and height.

Just as squares are specialized rectangles, so are cubes specialized rectangular solids. For a cube, the length, depth, and height are all equal.

The surface area of a solid is the area of its outermost skin. A cardboard box, for example, is made up of a bunch of rectangles fastened together. The sum of the areas of those rectangles is the surface area of the cardboard box.

To calculate the surface area of a rectangular solid, all you have to do is find the area of each of the sides and add them together. In fact, your job is even easier than that. The six sides of a rectangular solid can be divided into three pairs of two. If you look at the rectangular solid diagramed above, you should see that panel ABFE = DCGH, BCDA = FGHE, and BCGF = ADHE. Therefore, you only have to calculate the areas of one of each of the three pairs, sum those areas, and multiply that answer by 2.

With a cube, finding the surface area is even easier. By definition, each side of a cube will always be the same, so to calculate the surface area, find the area of one side, and multiply it by 6.

Finally, there is one property of surface area of which you should be aware. Pictured below is a rectangular solid that has a length of 8, a depth of 4, and a height of 4. Then a giant cleaver comes down and cuts the solid into two cubes, each of which have lengths, widths, and heights of 4. Do the two cubes have a bigger combined surface area? A smaller combined surface area? Or a combined surface area equal to the original solid? The answer is that the two cubes have a bigger surface area. Think about the cleaver coming down: it creates two new faces that weren’t there before.

Whenever something is cut into smaller pieces, its surface area increases (although its volume is unchanged). The SAT loves to test this fact.

The volume of a rectangular solid can be found by multiplying the length width height (V = lwh; this formula can be found in the reference area at the beginning of each math SAT section).

Because all the dimensions of the cube are equal, the volume of a cube is even easier to calculate: just raise the length of one edge to the third power. If a cube has a length, width, and height of 3, the volume is 3³ = 27.

You probably know what a cube or rectangular solid looks like, but you might not know what a right circular cylinder looks like. Here’s a picture of one:

You will never have to calculate the surface area of a right circular cylinder on the SAT.

It’s perfectly likely that you will have to calculate the volume of a right circular cylinder. Luckily, the formula isn’t so hard, and it’s available in the reference bar at the beginning of every SAT math section. Of course, as we say every time, you should still memorize the formula. The formula is: Volume = Like all other volumes, to get the volume of a right cylinder, you have to find the area of a base (in this case a circle) and then multiply it by the height.

Often, the SAT will pose its problems on solids as word problems. It will say something like: what is the total surface area of two boxes, each with dimensions of Often, the best approach to these word problems is to draw a sketch:

Once you see the drawing, you should see that you have four panels of four of and four of meaning that the total area is 4(15) + 4(20) + 4(12) = 60 + 80 + 48 = 188.

Whenever you see a word problem, it’s a good idea to draw a sketch.