ACT study guide Math: Perimeter, Area, Volume, and Surface Area

In inches, what is the radius of a sphere with a volume of 36π?

(The volume of any sphere is 4/3πr^3, where r is the radius. In this case, 43πr3=36π, and r^3 = 3/4(36) = 27. Finally, r, the cube root of 27, is )3

The perimeter of a square is 48 centimeters. What is its area, in square centimeters?

(If a square has side x, then its perimeter is 4x; this is because a square is defined as a rectangle where all four sides are of equal length. Since the perimeter of the square is 48, then 48 = 4x and x=48/4=12. Thus, the length of one side of the square is 12. The area of a square is defined as (side)2; therefore the area of this square is 122 or) 144

In square meters, what is the area of a single face of a cube that has a volume of 125 cubic meters?

(If s is a single side of the cube, s3 = 125, and s = 5. Therefore, a single face would have an area of 5 × 5 = )25

The perimeter of a square is 36 units. How many units long is the diagonal of the square?

(If the perimeter of a square is 36 units, then each side is 9 (since perimeter in a square is 4(s) where s is the length of a side). To find the length of the diagonal, you can use the Pythagorean Theorem because the diagonal is the hypotenuse of a right triangle with legs of length 9. Thus d2 = 92 + 92, or 81 + 81, which equals 2(81), and d=√(2(81)), or √2√81, which is) 9√2

The volume of a cube is given by the formula s^3, where s is the length of a side. If a cube has a volume of 64, and the length of each side is halved, the new cube's volume will be:

(In order to solve this problem, you must realize that since the volume of a cube is equal to the cube of its sides, multiplying the length of the sides by 1/2 will have the effect of multiplying the volume by (1/2)^3=18. The cube in this problem has a volume of 64, so if you halve the length of each side, new cube's volume will be 64(1/8)=)8

If the edges of a cube are tripled in length to produce a new, larger cube, then the larger cube's surface area is how many times larger than the smaller cube's surface area?

(Let the length of the edge of the smaller cube be s. The surface area is then 6s2. If the length of the edges are tripled, then s is replaced by 3s, making the surface area 6(3s)2 = (9) 6s2, or 9 times larger than the initial surface area.) 9

A square pool with an area of 81 square feet is to be placed entirely within a circular enclosure with a radius of 10 feet. Tiles will be laid within the entire enclosure around the pool (but not under it). What is the approximate area, in square feet, of the enclosure that will be tiled?

(Since the area of the square pool is given, you must find the area of the circle, with a radius of 10, and subtract the area of the pool. The area of a circle is equal to πr2, where r is the radius. The area of this circle is 102π = 100π ≈ 314 square feet. Thus the area of the enclosure is approximately 314 − 81 = )233 square feet.

The figure below shows the plan for the ground floor of a townhouse. The thickness of the walls should be ignored when answering the questions. The dimensions shown are in feet, and each region is rectangular. What is the perimeter, in feet, of the ground floor of the townhouse? whole townhouse ground floor width: 20 length: 18 townhouse ground floor kitchen (upper right) width: 8 length: 4 Town house ground floor dinning room door length: 4

(The perimeter is equal to the distance around an object. To calculate the perimeter of the ground floor of the townhouse, add the sides. The perimeter of the ground floor of the townhouse is 2(20′) + 2(18′), or 40′ + 36′, which is) 76

The figure above shows the plan for the ground floor of a townhouse. The thickness of the walls should be ignored when answering the questions. The dimensions shown are in feet, and each region is rectangular. What is the area, in square feet, of the living room? whole townhouse ground floor width: 20 length: 18 townhouse ground floor kitchen (upper right) width: 8 length: 4 Town house ground floor dinning room door length: 4

(To calculate the area of the living room, first calculate its dimensions. The length of the living room is 18′ less the width of the hallway, which is 4′, making it 18 − 4 = 14′. The width of the living room is 20′ less the width of the kitchen, which is 8′, making it 20 − 8 = 12′. Thus the area of the living room is 14′ × 12′ =) 168 square feet

After a snowstorm, city workers removed an estimated 12,000 cubic meters of snow from the downtown area. If this snow were spread in an even layer over an empty lot with dimensions 62 meters by 85 meters, about how many meters deep would the layer of snow be?

(To find the uniform depth, use the formula for volume, V, of a rectangular prism with the height h, length l, and width w, V = (l) (w) (h). Substitute the given values for the variables and solve for h: 12,000 = (62) (85) (h), or 12,000 = 5, 270h. Thus h=12,0005,270, or about 2.277, which is )between 2 and 3

If the volume of a cube is 64, what is the shortest distance from the center of the cube to the base of the cube?

(To solve this problem, remember that the volume of a cube is equal to (length)(width)(height) or simply (side)3, since all sides of a cube are equivalent in length. To find the length of one side, find the cube root of 64, which is 4 (43 = 64). Because all sides of a cube are equal, the shortest distance from the center of the cube to the base of the cube will equal the midpoint of the length of the cube, which is 42 , or )2

In feet, what is the perimeter of a square that has an area of 49 square feet?

If s is one side of the square, then s2 = 49, and s = 7. Therefore, the perimeter is 4(7) = 28

(This figure has 10 sides, but the lengths are given for only 7 sides. Those lengths add up to 36 inches. The perimeter is greater than this because of the missing 3 sides so you can eliminate answer choices A and B. To solve this problem, use the information given to find the missing sides; based on the figure, you can see that the sum of right-facing sides equals the sum of left-facing sides, and the sum of top-facing sides equals the sum of bottom-facing sides. It is easy to see that the bottom-facing sides will equal the top-facing side, which has a length of 14. Since we have the values for all of the left-facing sides (5 + 4 + 3 = 12), the right-facing sides also have the sum of 12. Thus the perimeter is 14 + 14 + 12 + 12, or )52

In the figure shown below, each pair of intersecting line segments meets at a right angle, and all the lengths are given in inches. What is the perimeter, in inches, of the figure?

First find the radius of the cone: V=1/3πr^2h 3V/πh = r^2 3(302)/π(8) = r^2 36=r^2 r=6 Then use the Pythagorean Theorem to find the slant height l: l^2=r^2+h^2 l^2=6^2+8^2=100 l=10

The volume of a right circular cone is determined by the following formula: V=1/3πr^2h If the volume of a certain cone is 302 cm3 and the height is 8 cm, determine the slant height of the cone

In order to clean her aquarium, Stephanie must remove half of the water. The aquarium measures 30 inches long, 16 inches wide, and 12 inches deep. The aquarium is currently completely full. What volume of water, in cubic inches, must Stephanie remove?

To solve this problem, calculate the volume of the aquarium and divide by 2. Since volume is equivalent to length × width × height, the volume is 30 × 16 × 12, or 5,760 cubic inches of water. Dividing by two, you see that half of the tank would be 2,880 cubic inches of water