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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

Area and Volume of Similar Solids

Two shapes are similar if all their corresponding angles are congruent and all their corresponding sides are proportional. Two solids are similar if they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional.

Surface Areas of Similar Solids

In two dimensions, when two shapes are similar, the ratio of their areas is the square of the scale factor. A comparable relationship holds in three dimensions as well.

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

Volumes of Similar Solids

Just like surface area, volumes of similar solids have a relationship that is related to the scale factor.

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

Summary

Ratios Units
Scale Factor \begin{align*}\frac{a}{b}\end{align*} \begin{align*}\text{in, ft, cm, m, etc.}\end{align*} 
Ratio of the Surface Areas \begin{align*}\left(\frac{a}{b}\right)^2\end{align*} \begin{align*}\mathrm{in^2 \! , \ ft^2 \!, \ cm^2 \!, \ m^2 \!, \ etc.}\end{align*}                           
Ratio of the Volumes \begin{align*}\left(\frac{a}{b}\right)^3\end{align*} \begin{align*}\mathrm{in^3 \! , \ ft^3 \!, \ cm^3 \!, \ m^3 \!, \ etc.}\end{align*}

Suppose you were given two similar square prisms and told the scale factor of their sides. How do you think you might find the ratio of their surface areas and the ratio of their volumes?

 

Examples

Example 1

Determine if the two triangular pyramids are similar.

Match up the corresponding parts.

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

These triangle pyramids are not similar.

Example 2

Two similar triangular prisms are below. The ratio of their volumes is 343 : 125. Find the missing sides in both triangles.

The scale factor is equal to 7:5, since that is the cube root of 343:125. With the scale factor, we can now set up several proportions.

\begin{align*}& \frac{7}{5} = \frac{7}{y} && \frac{7}{5}=\frac{x}{10} && \frac{7}{5}=\frac{35}{w} && 7^2+x^2=z^2 && \frac{7}{5}=\frac{z}{v}\\ & y = 5 && x=14 && w=25 && 7^2+14^2=z^2\\ & && && && z=\sqrt{245}=7\sqrt{5} && \frac{7}{5}=\frac{7\sqrt{5}}{v} \rightarrow v=5 \sqrt{5}\end{align*}

Example 3

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

Match up the corresponding heights, widths, and lengths.

\begin{align*}\frac{\mathrm{small \ prism}}{\mathrm{large \ prism}}: \frac{3}{4.5}=\frac{4}{6}=\frac{5}{7.5}\end{align*}

The congruent ratios tell us the two prisms are similar.

Example 4

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, \begin{align*}\sqrt{\frac{16}{25}}=\frac{4}{5}\end{align*}. Set up a proportion to find \begin{align*}h\end{align*}.

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

Example 5

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. \begin{align*}3^3:4^3 = 27:64\end{align*}.

Review

Determine if each pair of right solids are similar.

  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 \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. The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
  9. 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?
  10. The ratio of the surface areas of two cubes is 64:225. What is the ratio of the volumes?

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 15-18.

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

Use the hemispheres below to answer questions 19-20.

  1. Are the two hemispheres similar? How do you know?
  2. Find the ratio of the surface areas and volumes.
  1. The ratio of the surface areas of two similar cylinders is 16:81. What is the ratio of the volumes?

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.9. 

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Vocabulary

TermDefinition
similar solids Two solids that are the same type of solid and have corresponding radii, heights, base lengths, widths, etc. that are proportional.
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.

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