SAT Test Prep

CHAPTER 10
ESSENTIAL GEOMETRY SKILLS

Lesson 7: Volumes and 3-D Geometry

Volume


The SAT math section may include a question or two about volumes. Remember two things:

• The volume of a container is nothing more than the number of “unit cubes” it can hold.

• The only volume formulas you will need are given to you in the “Reference Information” on every math section.


Example:

How many rectangular bricks measuring 2 inches by 3 inches by 4 inches must be stacked together (without mortar or any other material) to create a solid rectangular box that measures 15 inches by 30 inches by 60 inches?

Don’t be too concerned with how the bricks could be stacked to make the box; there are many possible arrangements, but the arrangement doesn’t affect the answer. All you need to know is that it can be done. If so, just looking at the volumes is enough: if you use n bricks, then the box must have a volume that is n times larger than the volume of one brick. Each brick has a volume of  cubic inches. The box has a volume of  square inches. The number of bricks, then, must be .

3-D Distances


If you are trying to find the length of a line segment in three dimensions, look for a right triangle that has that segment as its hypotenuse.


Example:

The figure at right shows a cube with edges of length 4. If point C is the midpoint of edge BD, what is the length of 

Draw segment  to see that  is the hypotenuse of right triangle ΔAEC.

Leg  has a length of 4, and leg  is the hypotenuse of right triangle ΔEBC, with legs of length 2 and 4. Therefore,  


One possible shortcut for finding lengths in three dimensions is the three-dimensional distance formula:


If you think of point A in the cube above as being the origin (0, 0, 0), then point C can be considered to be (4, 4, 2). The distance from A to C, then, is

Concept Review 7: Volumes and 3-D Geometry

1. What is the definition of volume?

2. Write the formula for the volume of a rectangular box.

3. Write the 3-D distance formula.

4. Graph the points A (–2, 3, 1) and B (2, 1, (2) on an x-y-z graph.

5. What is the distance from point A to point B in the figure above?

6. The two containers with rectangular sides in the figure above have the interior dimensions shown. Both containers rest on a flat, horizontal surface. Container A is filled completely with water, and then this water is poured, without spilling, into Container B. When all of the liquid is poured from Container A into Container B, what is the depth of the water in Container B?

SAT Practice 7: Volumes and 3-D Geometry

1. The length, width, and height of a rectangular box, in centimeters, are a, b, and c, where a, b, and c are all integers. The total surface area of the box, in square centimeters, is s, and the volume of the box, in cubic centimeters, is v. Which of the following must be true?

I. v is an integer.

II. s is an even integer.

III. The greatest distance between any two vertices of the box is .

(A) I only

(B) I and II only

(C) I and III only

(D) I and III only

(E) I, II, and III

2. The figure above shows a rectangular box in which , and F is the midpoint of . What is the length of the shortest path from A to F that travels only on the edges of the box and does not pass through either point B or point C?

(A) 27.5

(B) 28.5

(C) 29.5

(D) 30

(E) 30.5

3. A pool-filling service charges $2.00 per cubic meter of water for the first 300 cubic meters and $1.50 per cubic meter of water after that. At this rate, how much would it cost to have the service fill a rectangular pool of uniform depth that is 2 meters deep, 20 meters long, and 15 meters wide?

(A) $450

(B) $650

(C) $800

(D) $1,050

(E) $1,200

4. In the figure above, a rectangular box has the dimensions shown. N is a vertex of the box, and M is the midpoint of an edge of the box. What is the length of ?

5. A cereal company sells oatmeal in two sizes of cylindrical containers. The smaller container holds 10 ounces of oatmeal. If the larger container has twice the radius of the smaller container and 1.5 times the height, how many ounces of oatmeal does the larger container hold? (The volume of a cylinder is given by the formula )

(A) 30

(B) 45

(C) 60

(D) 75

(E) 90

6. The figure above shows a rectangular solid with a volume of 72 cubic units. Base ABCD has an area of 12 square units. What is the area of rectangle ACEF?

7. The figure above shows a wedge-shaped holding tank that is partially filled with water. If the tank is 1/16 full, what is the depth of the water at the deepest part?

(A) 3

(B) 2

(C) 1.5

(D) 1

(E) 0.75

Answer Key 7: Volumes and 3-D Geometry

Concept Review 7

1. The volume of a solid is the number of “unit cubes” that fit inside of it.

4. Your graph should look like this one:

5. Using the 3-D distance formula,

6. Since the water is poured without spilling, the volume of water must remain the same. Container A has a volume of  cubic inches. Since Container B is larger, the water won’t fill it completely, but will fill it only to a depth of h inches. The volume of the water can then be calculated as  cubic inches. Since the volume must remain the same, , so  inches.

SAT Practice 7

1. E  so if a, b, and c are integers, v must be an integer also and statement I is true. The total surface area of the box, s, is , which is a multiple of 2 and therefore even. So statement II is true. Statement III is true by the 3-D distance formula.

2. C The path shown above is the shortest under the circumstances. The length of the path is

3. D The volume of the pool is  cubic meters. The first 300 cubic meters cost , and the other 300 cubic meters cost , for a total of $1,050.

4. C Draw segment  as shown. It is the hypotenuse of a right triangle, so you can find its length with the Pythagorean theorem:

 is the hypotenuse of right triangle NPM, so

5. C If the volume of the smaller container is , then the volume of the larger container is . So the larger container holds six times as much oatmeal as the smaller one. The smaller container holds 10 ounces of oatmeal, so the larger one holds  ounces.

6. 30 Mark up the diagram as shown. If the base has an area of 12, AB must be 4. If the volume of the box is 72, then the height must be AC must be 5, because it’s the hypotenuse of a 3-4-5 triangle. So the rectangle has an area of .

7. A If the volume of the water is 1/16 the volume of the tank, the smaller triangle must have an area 1/16 that of the larger triangle. The two are similar, so the ratio of the lengths must be 1/4, because . Therefore, the depth of water is 1/4 the depth of the tank: .