Unit 8 Lesson 12: Similar Figures and Volume

Question 1

cubed

Answer 1

a number multiplied by itself three times, indicated as a raised 3 on the right corner of the number or units

Question 2

dilation

Answer 2

a transformation technique that is used to make figures larger or smaller in size, without changing or distorting the shape

Question 3

scale factor

Answer 3

a number representing the ratio between the dimension of an object and the dimension of a dilation of that object

Question 4

volume

Answer 4

the amount of space occupied by a three-dimensional object as measured in cubic units

Question 5

Volume of triangular prism

Answer 5

bhl/2

Question 6

Volume of culinder

Answer 6

pi r^2h

Question 7

volume of cone

Answer 7

pi r^2h/3

Question 8

volume of pyramid

Answer 8

lwh/3

Question 9

volume of sphere

Answer 9

4/3 pi r^3

Question 10

volume of recangular prism

Answer 10

v = lxh

Question 11

A triangular prism has a volume of 8 in.^3
. Calculate the volume of a similar triangular prism if the scale factor is 3.

Answer 11

Solution

First, cube the scale factor: 33=27
.
Next, multiply by the given volume of the similar shape: 8⋅27=216
.
The triangular prism has a volume of 216 in.3
.

Alternatively, use the ratio of volumes based on the scale factor: The ratio of side lengths from the larger prism to the smaller prism is 3:1. Therefore, their volumes have a ratio of 33:1
. The larger prism has 27 times the volume of the smaller prism. 8⋅27=216 in.3
.

Question 12

Two rectangular prisms are similar in shape. The first rectangular prism has a volume of 16 cm^3
and the second rectangular prism has a volume of 1,024 cm^3
. Compare the volume of the two figures and determine the scale factor.

Answer 12

Solution

Compare the two volumes by dividing.
1,024÷16=64
; this is not the scale factor but the cube of the scale factor.
To find the true scale factor, calculate the cube root of 64.
64(13)=4
The scale factor is 4.
To check your answer, take the volume of the first figure and multiply it by the cubed scale factor.

(16)(43)=16⋅64=1,024

Question 14

Suppose the side lengths of figure B are 25% as long as those of similar figure A. What is the ratio of their side lengths, and what does it mean? What is the ratio of their volumes, and what does it mean?

Answer 14

The ratio of the side lengths of figure A to figure B is 4:1. All side lengths of figure A are 4 times as long as the corresponding sides of figure B.

The ratio of the volumes of figure A to figure B is 64:1. Since the side lengths differ by a factor of 4, the volumes differ by a factor of 4^3
. Figure A is 64 times as large as figure B.

Question 14

To find the volume of a similar shape, what must you do to the scale factor before multiplying it by the original volume?

Answer 14

You must cube the scale factor.

Question 15

cone

Answer 15

a solid bounded by a circular or other closed plane base and the surface formed by line segments joining every point of the boundary of the base to a common vertex

Question 16

Therefore, the dilation of the surface area and volume of a three-dimensional figure varies depending on the scale factor. Review how scale factor affects dilation:

Answer 16

• Perimeter: dilation is a single measurement.
• Example: A scale factor of 2 would be calculated as 2 times the length.
• Surface Area: dilation is a squared measurement.
• Example: A scale factor of 2 would be calculated as 22
• , or 4 times the size in area.
• Volume: dilation is a cubic measurement.
• Example: A scale factor of 2 would be calculated as 23
• , or 8 times the size in volume.

Question 17

To determine the new volume of a figure that has a dilation, follow the steps below.

Answer 17

• First, find the volume of the original figure.
• Next, cube the scale factor of dilation.
• Last, multiply the original volume by the cubed scale factor.

Question 18

Does volume grow at the same rate as surface area when a three-dimensional figure is dilated? Explain your reasoning.

Answer 18

No, volume grows or shrinks as a cubed rate, while surface area is a squared rate to the indicated scale factor.

Question 19

One cube has side length 1.52, and a second cube has side length 2.89. Use ratios to compare the side lengths and volumes of the cubes.

Answer 19

The ratio of side lengths from the large cube to the small cube is 2.891.52≈1.901
. The side lengths of the large cube are about twice as long as those of the small cube.

Sample answer: The ratio of volumes from the large cube to the small cube is (2.891.52)3≈6.871
. The volume of the large cube is about 7 times that of the small cube.

Question 20

A cone has a volume capacity of 405 inches3
. If the cone was dilated using a scale factor of 13
, what would be the new volume capacity of the cone?

Answer 20

15 inches3

First, the volume of the original figure is V=405 inches3.

Next, cube the scale factor.

(13)3=(13)(13)(13)=127

Last, multiply the original volume by the cubed scale factor.

405⋅127=15

The volume of the new cone is 15 inches3
. The original volume of the cone went down by a factor of 127
.

Question 21

If a cone has a surface dilation of 16, what would be its volume dilation?

Answer 21

If a surface dilation is 16, that means the scale factor is 4, because the surface area dilation of any figure is a squared scale factor. Since the volume dilation of a figure is a cubic scale factor, the volume dilation of the cone is (4)3=(4)(4)(4)=64
.