11.7: Exploring Similar Solids
Learning Objectives
- Find the relationship between similar solids and their surface areas and volumes.
Review Queue
- We know that every circle is similar, is every sphere similar?
- Find the volume of a sphere with a 12 in radius. Leave your answer in terms of \begin{align*}\pi\end{align*}.
- Find the volume of a sphere with a 3 in radius. Leave your answer in terms of \begin{align*}\pi\end{align*}.
- Find the scale factor of the spheres from #2 and #3. Then find the ratio of the volumes and reduce it. What do you notice?
- Two squares have a scale factor of 2:3. What is the ratio of their areas?
- The smaller square from #5 has an area of \begin{align*}16 \ cm^2\end{align*}. What is the area of the larger square?
- The ratio of the areas of two similar triangles is 1:25. The height of the larger triangle is 20 cm, what is the height of the smaller triangle?
Know What? Your mom and dad have cylindrical coffee mugs with the dimensions to the right. Are the mugs similar? (You may ignore the handles.) If the mugs are similar, find the volume of each, the scale factor and the ratio of the volumes.
Similar Solids
Recall that two shapes are similar if all the corresponding angles are congruent and the corresponding sides are proportional.
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.
Example 1: Are the two rectangular prisms similar? How do you know?
Solution: Match up the corresponding heights, widths, and lengths to see if the rectangular prisms are proportional.
\begin{align*}\frac{small \ prism}{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 2: Determine if the two triangular pyramids similar.
Solution: Just like Example 1, let’s match up the corresponding parts.
\begin{align*}\frac{6}{8}=\frac{12}{16}=\frac{3}{4}\end{align*} however, \begin{align*}\frac{8}{12}=\frac{2}{3}.\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.
Surface Areas of Similar Solids
Recall that when two shapes are similar, the ratio of the area is a square of the scale factor.
For example, the two rectangles to the left are similar because their sides are in a ratio of 5:8. The area of the larger rectangle is \begin{align*}8(16)=128 \ units^2\end{align*} and the area of the smaller rectangle is \begin{align*}5(10)=50 \ units^2\end{align*}. If we compare the areas in a ratio, it is \begin{align*}50:128=25:64= 5^2=8^2\end{align*}.
So, what happens with the surface areas of two similar solids? Let’s look at Example 1 again.
Example 3: Find the surface area of the two similar rectangular prisms.
Solution:
\begin{align*}SA_{smaller}&=2(4 \cdot 3)+2(4 \cdot 5)+2(3 \cdot 5)\\ & = 24+40+30=94 \ units^2\end{align*}
\begin{align*}SA_{larger}&=2(6 \cdot 4.5)+2(4.5 \cdot 7.5)+2(6 \cdot 7.5)\\ & = 54+67.5+90=211.5 \ units^2\end{align*}
Now, find the ratio of the areas. \begin{align*}\frac{94}{211.5}=\frac{4}{9}=\frac{2^2}{3^2}\end{align*}. The sides are in a ratio of \begin{align*}\frac{4}{6}=\frac{2}{3}\end{align*}, so the surface areas have the same relationship as the areas of two similar shapes.
Surface Area Ratio: If two solids are similar with a scale factor of \begin{align*}\frac{a}{b}\end{align*}, then the surface areas are in a ratio of \begin{align*}\left ( \frac{a}{b} \right )^2\end{align*}.
Example 4: Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller cylinder?
Solution: 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*}. Now we can set up a proportion to find \begin{align*}h\end{align*}.
\begin{align*}\frac{4}{5} & = \frac{24}{h}\\ 4h&=120\\ h&=30\end{align*}
Example 5: Using the cylinders from Example 4, if the surface area of the smaller cylinder is \begin{align*}1536 \pi \ cm^2\end{align*}, what is the surface area of the larger cylinder?
Solution: Set up a proportion using the ratio of the areas, 16:25.
\begin{align*}\frac{16}{25}&=\frac{1536 \pi}{A}\\ 16A&=38400 \pi \\ A&=2400 \pi \ cm^2\end{align*}
Volumes of Similar Solids
Let’s look at what we know about similar solids so far.
Ratios | Units | |
---|---|---|
Scale Factor | \begin{align*}\frac{a}{b}\end{align*} | in, ft, cm, m, etc. |
Ratio of the Surface Areas | \begin{align*}\left ( \frac{a}{b} \right )^2\end{align*} | \begin{align*}in^2, ft^2, cm^2, m^2,\end{align*} etc. |
Ratio of the Volumes | ?? | \begin{align*}in^3, ft^3, cm^3, m^3,\end{align*} etc. |
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. We will do an example and test our theory.
Example 6: Find the volume of the following rectangular prisms. Then, find the ratio of the volumes.
Solution:
\begin{align*}V_{smaller}&=3(4)(5)=60\\ V_{larger}&=4.5(6)(7.5)=202.5\end{align*}
The ratio is \begin{align*}\frac{60}{202.5}\end{align*}, which reduces to \begin{align*}\frac{8}{27}=\frac{2^3}{3^3}\end{align*}.
It seems as though our prediction based on the patterns is correct.
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*}.
Example 7: Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes?
Solution: If we cube 3 and 4, we will have the ratio of the volumes. Therefore, \begin{align*}3^3:4^3\end{align*} or 27:64 is the ratio of the volumes.
Example 8: If the ratio of the volumes of two similar prisms is 125:8, what is their scale factor?
Solution: This example is the opposite of the previous example. We need to take the cubed root of 125 and 8 to find the scale factor.
\begin{align*}\sqrt[3]{125}:\sqrt[3]{8}=5:2\end{align*}
Example 9: Two similar right triangle prisms are below. If the ratio of the volumes is 343:125, find the missing sides in both figures.
Solution: 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.
\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 10: The ratio of the surface areas of two similar cylinders is 16:81. If the volume of the smaller cylinder is \begin{align*}96 \pi \ in^3\end{align*}, what is the volume of the larger cylinder?
Solution: 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, we need cube this to find the ratio of the volumes, \begin{align*}4^3:9^3 = 64:729\end{align*}. At this point we can set up a proportion to solve for the volume of the larger cylinder.
\begin{align*}\frac{64}{729}&= \frac{96 \pi}{V}\\ 64V&=69984 \pi\\ V& =1093.5 \pi \ in^3\end{align*}
Know What? 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 \begin{align*}54 \pi \ in^3\end{align*} and Mom’s mug is \begin{align*}16 \pi \ in^3\end{align*}. The ratio of the volumes is \begin{align*}54 \pi : 16 \pi\end{align*}, which reduces to 8:27.
Review Questions
Determine if each pair of right solids are similar. Explain your reasoning.
- Are all cubes similar? Why or why not?
- Two prisms have a scale factor of 1:4. What is the ratio of their surface areas?
- Two pyramids have a scale factor of 2:7. What is the ratio of their volumes?
- Two spheres have radii of 5 and 9. What is the ratio of their volumes?
- The surface area of two similar cones is in a ratio of 64:121. What is the scale factor?
- The volume of two hemispheres is in a ratio of 125:1728. What is the scale factor?
- 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?
- A cube has sides of length \begin{align*}x\end{align*} and is enlarged so that the sides are \begin{align*}4x\end{align*}. How does the volume change?
- The ratio of the volumes of two similar pyramids is 8:27. What is the ratio of their total surface areas?
- 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?
- The ratio of the surface areas of two cubes is 64:225. If the volume of the smaller cube is \begin{align*}13824 \ 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.
- What is the scale factor?
- What is the ratio of the surface areas?
- Find \begin{align*}h, x\end{align*} and \begin{align*}y\end{align*}.
- Find \begin{align*}w\end{align*} and \begin{align*}z\end{align*}.
- Find the volume of both pyramids.
- Find the lateral surface area of both pyramids.
Use the hemispheres below to answer questions 22-25.
- Are the two hemispheres similar? How do you know?
- Find the ratio of the surface areas and volumes.
- Find the lateral surface areas of both hemispheres.
- Determine the ratio of the lateral surface areas for the hemispheres. Is it the same as the ratio of the total surface area? Why or why not?
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.
- Find the ratio of the strengths of the bones. How much stronger are the bones in animal B?
- If their weights are proportional to their volumes, find the ratio of their weights.
- Which animal has a skeleton more capable of supporting its own weight? Explain.
Two sizes of cans of beans are similar. The thickness of the walls and bases are the same in both cans. The ratio of their surface areas is 4:9.
- If the surface area of the smaller can is 36 sq in, what is the surface area of the larger can?
- If the sheet metal used to make the cans costs $0.006 per square inch, how much does it cost to make each can?
- What is the ratio of their volumes?
- If the smaller can is sold for $0.85 and the larger can is sold for $2.50, which is a better deal?
Review Queue Answers
- Yes, every sphere is similar because the similarity only depends on one length, the radius.
- \begin{align*}\frac{4}{3} 12^3 \pi = 2304 \pi \ in^3\end{align*}
- \begin{align*}\frac{4}{3} 3^3 \pi = 27 \pi \ in^3\end{align*}
- The scale factor is 4:1, the volume ratio is 2304:36 or 64:1
- \begin{align*}\frac{4}{9}\end{align*}
- \begin{align*}\frac{4}{9} = \frac{16}{A} \rightarrow A = 36 \ cm^2\end{align*}
- \begin{align*}\frac{1}{5} = \frac{x}{20} \rightarrow x = 4 \ cm\end{align*}