Solids

Words of “Wiz-dom”-You’ll see two possible shapes of solids that are treated as solids on the test. The most common is a rectangular (or cubic) solid. The formula for its volume is in the SAT “Cheat Sheet” at the beginning of the math section. Keep in mind that its surface area and perimeters of individual faces are the same as rectangles and squares. The less common type is a right cylinder. That’s a can! They are simply a pair of identical circles connected by a rectangle that has been rolled into a tube. Prove it to yourself by cutting a paper towel or toilet paper roll. You’ll also find the formula for the volume at the beginning of an SAT math section.  If you are taking the ACT, be sure to memorize these formulas.

Having said there are two possible shapes that are treated as solids, the test writer has been known to sneak cones and pyramids onto the test. Don’t panic. Relax. The questions will be about the circle that is the base of the cone (so it’s a circle question in disguise) or a proportion problem. Since 1991, I don’t remember seeing a pyramid question in which the base wasn’t a square.  I’ve never seen a volume or surface area question that involves the side of the cone or square. We’ll do some sample questions to ease your concerns.

Let’s deal with these solids one at a time.

Rectangular Solids

The following questions relate to the solid for which the height is 3, the length is 6, and the width is 4.

What is the volume of the solid? Answer #1

What is the surface area of the solid? Answer #2

What is the length of the diagonal AB of the end? Answer #3

What is the perimeter of the top (or bottom)? Answer #4

What is the ratio of the height to the length? Answer #5

If an ant walks only on the edges, what is the length of the shortest route from A to C? Answer #6

 Right Cylinders

The following questions relate to the solid for which the height is 8 and the radius is 3.

What is the volume? Answer #7

What is the surface area? Answer #8

What is the circumference? Answer #9

What is the shortest distance from A to B (points exactly opposite one another?) Answer #10

If an ant runs around the top rim, then straight down the side, then runs around the bottom rim, how far did she go? Answer #11

Cones and Pyramids

    

Words of “Wiz-dom”—You won’t need to worry about surface area or volume unless your opponent includes the formula in the question.  They are almost always, if not always, about right triangles and frequently similar right triangles.  Begin by imagining a line from the peak straight down to the center of the base.  Then draw a line to the edge of the base of the cone.  In a pyramid, make the right triangle by drawing a line from the center of the base to a corner or the middle of one edge.

Sample Questions

5.  If the length of the side of a cube is doubled, what effect does it have on the surface area of the cube?

(A) 2 times as large
(B) 4 times as large
(C) 12 times as large
(D) 24 times as large
(E) 48 times as large

Answer to Question #5

15.  If the height of a right cylinder is tripled and its radius is doubled, how much larger is the volume?

(A) 5 times larger
(B) 6 times larger
(C) 12 times larger
(D) 16 times larger
(E) 24 times larger

Answer to Question #15

6.  A painter is going to paint the walls of The Wiz’s cogitating room that is 12 feet wide and 18 feet long. The walls are 8 feet high. He can cover 400 square feet of surface area with each gallon of paint. If he can only buy whole gallons of paint, how many gallons does he need to purchase?

(A) 5
(B) 4
(C) 3
(D) 2
(E) 1

Answer to Question #6

8.  What is the most number of small cubes with an edge of 15 that will fit in a large cube with an edge of 75?

(A) 250
(B) 125
(C) 100
(D) 25
(E) 5

Answer to Question #8

13.  A cube has a volume of 27. What is its surface area?

Answer to Question #13

12.  A rectangular solid has a length of 12, a height of 8, and a width of 6. What is the minimum length of tape needed to cover all of the edges without overlap?

Answer to Question #12

17.  A 12-inch tall cone with a base area of 81 is sliced into two pieces parallel to the base.  How tall is the new smaller cone if its base has an area of 36?

Answer to Question #17

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