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# Area and Volume of Similar Solids

## Solve problems using ratios between similar solids.

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Area and Volume of Similar Solids

Consider the cylindrical coffee mugs with different dimensions, pictured below. Are the mugs similar, aside from the handles? If the mugs are similar, would you know how to find the volume of each, the scale factor and the ratio of the volumes?

### Area and Volume of Similar Solids

Two solids are similar if and only if they are the same type of solid and their corresponding linear measures (radii, heights, base lengths, etc.) are proportional.

##### Surface Area

Recall that when two shapes are similar, the ratio of the area is the square of the scale factor.

For example, the two rectangles above are similar because their sides are in a ratio of 5:8. The area of the larger rectangle is 8(16)=128 units2\begin{align*}8(16)=128 \ \mathrm{units}^2\end{align*} and the area of the smaller rectangle is 5(10)=50 units2.\begin{align*}5(10)=50 \ \mathrm{units}^2.\end{align*} If we compare the areas in a ratio, it is 50:128 = 25:64 = 52:82.\begin{align*}50:128 \ = \ 25:64 \ = \ 5^2:8^2.\end{align*}

Surface Area Ratio: If two solids are similar with a scale factor of ab\begin{align*}\frac{a}{b}\end{align*}, then the surface areas are in a ratio of (ab)2.\begin{align*}\left ( \frac{a}{b} \right )^2.\end{align*}

##### Volume

Let’s look at what we know about similar solids so far.

Ratios Units
Scale Factor ab\begin{align*}\frac{a}{b}\end{align*} in,ft,cm,m,etc.\begin{align*}\mathrm{in, ft, cm, m, etc.}\end{align*}
Ratio of the Surface Areas (ab)2\begin{align*}\left ( \frac{a}{b} \right )^2\end{align*} in2, ft2, cm2, m2, etc.\begin{align*}\mathrm{in^2 \!, \ ft^2 \!, \ cm^2 \!, \ m^2 \!, \ etc.}\end{align*}
Ratio of the Volumes ?? in3, ft3, cm3, m3, etc.\begin{align*}\mathrm{in^3 \!, \ ft^3 \!, \ cm^3 \!, \ m^3 \!, \ etc.}\end{align*}

It looks as though there is a pattern. If the ratio of the volumes follows the pattern from above, it should be the cube of the scale factor.

Volume Ratio: If two solids are similar, with a scale factor of ab\begin{align*}\frac{a}{b}\end{align*}, then the volumes are in a ratio of (ab)3.\begin{align*}\left ( \frac{a}{b} \right )^3 \!.\end{align*}

#### Determining Similarity

1. Are the two rectangular prisms similar? How do you know?

Match up the corresponding heights, widths, and lengths to see if the rectangular prisms are proportional.

small prismlarge prism=34.5=46=57.5\begin{align*}\frac{\mathrm{small \ prism}}{\mathrm{large \ prism}}=\frac{3}{4.5}=\frac{4}{6}=\frac{5}{7.5}\end{align*}

2. The congruent ratios tell us the two prisms are similar.

Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller cylinder?

First, we need to take the square root of the area ratio to find the scale factor, 1625=45\begin{align*}\sqrt{\frac{16}{25}}=\frac{4}{5}\end{align*}. Now we can set up a proportion to find h\begin{align*}h\end{align*}.

454hh=24h=120=30\begin{align*}\frac{4}{5} & = \frac{24}{h}\\ 4h&=120\\ h&=30\end{align*}

#### Finding the Ratio of Volumes

Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes?

If we cube 3 and 4, we will have the ratio of the volumes. Therefore, 33:43\begin{align*}3^3:4^3\end{align*} or 27:64 is the ratio of the volumes.

#### Coffee Mug Problem Revisited

The coffee mugs are similar because the heights and radii are in a ratio of 2:3, which is also their scale factor. The volume of Dad’s mug is 54π in3,\begin{align*}54 \pi \ \mathrm{in}^3 \!,\end{align*} and Mom’s mug is 16π in3.\begin{align*}16 \pi \ \mathrm{in}^3 \!.\end{align*} The ratio of the volumes is 54π:16π\begin{align*}54 \pi : 16 \pi\end{align*}, which reduces to 8:27.

### Examples

#### Example 1

Determine if the two triangular pyramids are similar.

Let’s match up the corresponding parts.

68=1216=34\begin{align*}\frac{6}{8}=\frac{12}{16}=\frac{3}{4}\end{align*} however, 812=23  34.\begin{align*}\frac{8}{12}=\frac{2}{3} \ \neq \ \frac{3}{4}.\end{align*}

Because one of the base lengths is not in the same proportion as the other two lengths, these right triangle pyramids are not similar.

#### Example 2

If the ratio of the volumes of two similar prisms is 125:8, what is their scale factor?

We need to take the cubed root of 125 and 8 to find the scale factor.

1253:83=5:2\begin{align*}\sqrt[3]{125}:\sqrt[3]{8}=5:2\end{align*}

#### Example 3

Two similar right triangle prisms are below. If the ratio of the volumes is 343:125, find the missing sides in both figures.

If the ratio of the volumes is 343:125, then the scale factor is 7:5, the cubed root of each. With the scale factor, we can now set up several proportions.

75=7y75=x1075=35w72+x2=z275=zvy=5 x=14 w=25 72+142=z2 z=245=7575=75vv=55\begin{align*}& \frac{7}{5} = \frac{7}{y} \qquad \frac{7}{5}=\frac{x}{10} \qquad \frac{7}{5}=\frac{35}{w} \qquad 7^2+x^2=z^2 \qquad \qquad \frac{7}{5}=\frac{z}{v}\\ & y = 5 \qquad \ x=14 \qquad \ w=25 \qquad \ 7^2+14^2=z^2 \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ z=\sqrt{245}=7\sqrt{5} \qquad \frac{7}{5}=\frac{7\sqrt{5}}{v} \rightarrow v=5 \sqrt{5}\end{align*}

#### Example 4

The ratio of the surface areas of two similar cylinders is 16:81. If the volume of the smaller cylinder is 96π in3,\begin{align*}96 \pi \ \mathrm{in}^3 \!,\end{align*} what is the volume of the larger cylinder?

First we need to find the scale factor from the ratio of the surface areas. If we take the square root of both numbers, we have that the ratio is 4:9. Now, cube this to find the ratio of the volumes, 43:93=64:729\begin{align*}4^3:9^3 = 64:729\end{align*}. Now set up a proportion to solve for the volume of the larger cylinder:

6472964VV=96πV=69,984π=1093.5π in3\begin{align*}\frac{64}{729}&= \frac{96 \pi}{V}\\ 64V&=69, \!984 \pi\\ V& =1093.5 \pi \ \mathrm{in}^3\end{align*}

### Review

Determine if each pair of right solids are similar. Explain your reasoning.

1. Are all cubes similar? Why or why not?
2. Two prisms have a scale factor of 1:4. What is the ratio of their surface areas?
3. Two pyramids have a scale factor of 2:7. What is the ratio of their volumes?
4. Two spheres have radii of 5 and 9. What is the ratio of their volumes?
5. The surface area of two similar cones is in a ratio of 64:121. What is the scale factor?
6. The volume of two hemispheres is in a ratio of 125:1728. What is the scale factor?
7. A cone has a volume of 15π\begin{align*}15 \pi\end{align*} and is similar to another larger cone. If the scale factor is 5:9, what is the volume of the larger cone?
8. A cube has sides of length x\begin{align*}x\end{align*} and is enlarged so that the sides are 4x\begin{align*}4x\end{align*}. How does the volume change?
9. The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
10. The ratio of the volumes of two tetrahedrons is 1000:1. The smaller tetrahedron has a side of length 6 cm. What is the side length of the larger tetrahedron?
11. The ratio of the surface areas of two cubes is 64:225. If the volume of the smaller cube is 13,824 m3\begin{align*}13, \!824 \ \mathrm{m}^3\end{align*}, what is the volume of the larger cube?

Below are two similar square pyramids with a volume ratio of 8:27. The base lengths are equal to the heights. Use this to answer questions 16-21.

1. What is the scale factor?
2. What is the ratio of the surface areas?
3. Find h,x\begin{align*}h, x\end{align*} and y\begin{align*}y\end{align*}.
4. Find w\begin{align*}w\end{align*} and z\begin{align*}z\end{align*}.
5. Find the volume of both pyramids.
6. Find the lateral surface area of both pyramids.

Animal A and animal B are similar (meaning the size and shape of their bones and bodies are similar) and the strength of their respective bones are proportional to the cross sectional area of their bones. Answer the following questions given that the ratio of the height of animal A to the height of animal B is 3:5. You may assume the lengths of their bones are in the same ratio.

1. Find the ratio of the strengths of the bones. How much stronger are the bones in animal B?
2. If their weights are proportional to their volumes, find the ratio of their weights.

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### Vocabulary Language: English

TermDefinition
Area Area is the space within the perimeter of a two-dimensional figure.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.