# 9.10 Similarity in solid figures

Worksheet
Similarity in solid figures
1

Consider the rectangular prisms A and B shown.

a

Find the surface area of:

i

A

ii

B

b

Determine the ratio of the surface area of Rectangular Prism A to Rectangular Prism B?

c

If the matching sides of two similar figures are in the ratio m:3, find the ratio of their surface areas.

2

Two similar cones have matching heights and radii whose lengths are in the ratio 7:2. Find the ratio of their surface areas.

3

Two similar pyramids have matching sides in the ratio 2:5. If the surface area of the larger pyramid is 250\text{ cm}^2, find the surface area of the smaller pyramid.

4

Consider the two similar spheres shown. The smaller sphere has radius 3 \text{ cm} while the larger sphere has a radius of 12\text{ cm}.

a

Find the volume of Sphere A, in simplest exact form.

b

Find the volume of Sphere B, in simplest exact form.

c

Determine the ratio of the volume of Sphere A to Sphere B.

d

If the matching dimensions of two similar figures are in the ratio m\,:\,4, find the ration of their volumes.

5

Two similar cones have corresponding dimensions in the ratio 1:4. Find the ratio of their volumes.

6

Two similar pyramids have their surface areas in the ratio 16:25. If the volume of the smaller pyramid is 256\text{ cm}^3, find the volume of the larger pyramid.

7

Cosider the following two similar cylinders:

a

Find the fully simplified ratio of the volume of the larger cylinder to the volume of the smaller cylinder.

b

Using the answer to part (a), find the ratio of the height of the larger cylinder to the height of the smaller cylinder.

c

Using your answers to the previous two parts, find the value of c.

8

The following two solids are similar.

Find the value of x.

9

A miniature model of a tower is made with a ratio of1:20.

a

If the scaled down model required 50 \text{ kg} of steel, how much steel is required to build the real tower?

b

If it took 5 \text{ L} of paint to give one coat to the model, how much paint is required for one coat of the real tower?

10

Two similar square pyramids have base edges of length 5\text{ cm} and 20\text{ cm}.

a

Find the ratio of smaller surface areas to the larger surface area

b

Find the ratio of the corresponding volumes (smaller to larger).

c

If the larger pyramid has a surface area of 44 \text{ cm}^2, find the surface area of the smaller pyramid correct to two decimal places.

d

If the smaller pyramid has a volume of 60\text{ cm}^3, find the volume of the larger pyramid.

11

Consider the formula A = \dfrac{1}{2} b h. State the effect of each of the following on A.

a

The values of both b and h double.

b

The value of b increases by a factor of 2 and the value of h is decreased by a factor of 2.

12

Consider the formula A = \pi r^{2}. Determine the effect of the following conditions on A, what effect would this have on?

a

The value of r were to triple

b

r was decreased by a factor of 4

13

Susana has two teddy bears that have the same shape but are different sizes. The length of the first teddy bear is 15\text{ cm}, while the length of the second teddy bear is 75\text{ cm}.

a

Find the length scale factor.

b

Find the surface area scale factor.

c

If the smaller bear needs 375 \text{ cm}^2 to be covered with 'fake fur', find the needed 'fake fur' for the bigger bear.

14

Two similar cones have bases with radius 7\text{ cm} and 28\text{ cm} respectively.

a

Find the scale factor from the height of the smaller cone to the height of the larger cone.

b

Find the scale factor from the volume of the smaller cone to the volume of the larger cone.

c

Find the volume of the larger cone, if the volume of the smaller cone is 852\text{ cm}^3.

15

The radii of two spherical balloons are 12\text{ cm} and 6\text{ cm} respectively.

a

Indicate the ratio of the radii.

b

Find the ratio of their surface areas.

c

Find the ratio of their volumes.

d

Find the ratio of the their volumes, if half the air is released from the smaller balloon.

16

The surface areas of two similar triangular prisms are in the ratio 64:49.

a

Find the scale factor of their sides.

b

Find the scale factor of their volumes.

17

The volume of two similar crates are in the ratio 1331:125.

a

Find the ratio of their sides.

b

Find the ratio of their surface areas.

18

Two pentagonal prisms are similar. The areas of their cross-sectional faces are given.

a

Find the surface area scale factor of Figure II compared to Figure I.

b

Find the length scale factor.

c

Find the height of Figure II, if Figure I is 13\text{ mm} high.

19

Consider these two similar rectangular prisms:

a

Find the length scale factor.

b

Find the surface area scale factor.

c

Find the volume scale factor.

d

Find the volume scale factor, if the measurements of the smaller prism are doubled.

20

Sophia was making a practice birthday cake for her son. The dimensions are 8\text{ cm} for the length, 3\text{ cm} for the width and 3\text{ cm} for the height. The actual cake will have dimensions 24\text{ cm} for the length, 9\text{ cm} for the width and 9\text{ cm} for the height.

a

Find the ratio of the dimensions of the actual cake to those of the practice cake.

b

Find the ratio of the volume of the actual cake to that of the practice cake.

c

To make the actual cake, determine the number of times the quantities of the ingredients of the practice cake Sophia should use.

21

Consider the two similar trapezoidal prisms.

a

Find the length scale factor, going from the smaller prism to the larger prism.

b

Find the volume scale factor, going from the smaller prism to the larger prism.

22

The model of Eiffel tower is made with a ratio 1:6480.

a

Find the height of the model, in \text{cm}, if the height of Eiffel tower is 324 \text{ m}.

b

Find the ratio of the surface area of the model to the surface area of Eiffel tower.

23

The ratio of the length of a model car to a real car is 1:20.

a

Find the ratio of surface area of the model car to the real car.

b

Find the ratio of the volume of the model car to the real car.

c

Find how many liters of paint are needed to paint the real car, if 18 \text{ mL} are needed to paint the model car.

d

Find the capacity of the model car fuel tank, in \text{mL}, if the real car fuel tank holds 48 \text{ L}.

24

A beehive consists of hexagonal cells with side length 3.2\text{ mm} and depth of 3.8\text{ mm}.

A plastic container is built which is modelled on these hexagonal cells, with a side length of 6.4\text{ cm}.

a

Find the ratio of the length of the side of the beehive to the length side of the modelled storage unit.

b

Find the ratio of the surface area of the side of the beehive to the surface area side of the modelled storage unit.

c

Find the ratio of the volume of the beehive to the volume of the modelled storage unit.

25

The volume of a soup can is reduced from 125\text{ cm}^3 to 91.125\text{ cm}^3.

What is the scale factor?

26

An ice cream cone has a surface area of 20\text{ cm}^2. If the size of the cone is increased by a scale factor of 3.1, what will its new surface area be?

27

A geologist found two pieces of pyrite, or fool's gold. She found that the volume of Specimen B is 3.375 times greater than the volume of Specimen A.

a

What is the scale factor?

b

How many times greater is the surface area of Specimen B than Specimen A?

28

A regular pentagon has been enlarged with a scale factor of 3.25. Describe the relationship between the perimeter of the two pentagons.

29

Describe the relationship between the volume of the two cylinders shown.

30

Sanford Seminole Aquatic Center has a swimming pool that is 25 yards by 55 yards, and 2.5 yards deep. Morcom Aquatic Center has a swimming pool with a volume of 6187.50 cubic yards.

a

Identify the pool having the greater volume.

b

How many times bigger is the volume of the bigger pool than the smaller pool?

c

Determine the scale factor between the pool at Morcom and the pool at Sanford Seminole.

31

The volume of a regulation adult beach volleyball is 106\pi\text{ in}^3. The volume of a regulation youth volleyball is \dfrac{256}{3}\pi \text{ in}^3.

Determine the approximate ratio of the radius of the youth volleyball to the adult beach volleyball.

32

The Super Cereal Company sells their Frosty O's cereal in a rectangular box with dimensions 12\text{ in} \times 10 \text{ in} \times 2.5 \text{ in}. In order to reduce their environmental impact, Super Cereal wants to reduce the surface area of their boxes while maintaining the proportions of the current packaging.

a

List three different scale factors that Super Cereal could use to reduce the total surface area of Frosty O's boxes by somewhere between 15\text{ in}^2 and 25 \text{ in}^2.

b
State the dimensions for a Frosty O's box that satisfies the required reduction in surface area.
33

The Soso Cereal Company sells their Frosty Squares cereal in a rectangular box with dimensions 12 \text{ in} \times 10 \text{ in} \times 2.5 \text{ in}. In order to increase their profits, Soso Cereal wants to reduce the volume of their boxes, while maintaining the proportions of the current packaging.

a

List three different scale factors that Soso Cereal could use to reduce the total volume of Frosty Squares boxes by somewhere between 15\text{ in}^3 and 25 \text{ in}^3.

b
State the dimensions for a Frosty Squares box that satisfies the required reduction in volume.
34

Cylinder Y has radius r and height h. Cylinder Z has radius 5r and height \dfrac{1}{4}h. Write an equation that describes the relationship between the volume of Cylinder Y and the volume of Cylinder Z.

### Outcomes

#### G.14a

Apply the concepts of similarity to two- or three-dimensional geometric figuresspecifically, compare ratios between lengths, perimeters, areas, and volumes of similar figures

#### G.14b

Apply the concepts of similarity to two- or three-dimensional geometric figuresspecifically, determine how changes in one or more dimensions of a figure affect area and/or volume of the figure

#### G.14c

Apply the concepts of similarity to two- or three-dimensional geometric figuresspecifically, determine how changes in area and/or volume of a figure affect one or more dimensions of the figure

#### G.14d

Apply the concepts of similarity to two- or three-dimensional geometric figuresspecifically, solve problems, including practical problems, about similar geometric figures