What if you had to compare your parents' cylindrical coffee mugs with different dimensions, pictured below? 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. After completing this Concept, you'll be able to solve problems like these.
Watch This
CK-12 Foundation: Chapter11AreaanVolumeofSimilarSolidsA
Learn more about similarity and volume ratios by watching the video at this link.
Guidance
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 a 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 \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*}.
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*}.
Volume
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.
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 A
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.
\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 B
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*}. 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 C
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, \begin{align*}3^3:4^3\end{align*} or 27:64 is the ratio of the volumes.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter11AreaandVolumeofSimilarSolidsB
Concept 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 \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.
Vocabulary
Two solids are similar if they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional.
Guided Practice
1. Determine if the two triangular pyramids are similar.
2. If the ratio of the volumes of two similar prisms is 125:8, what is their scale factor?
3. Two similar right triangle prisms are below. If the ratio of the volumes is 343:125, find the missing sides in both figures.
4. 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?
Answers:
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.
2. 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*}
3. 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*}
4. 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*}
Practice
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.
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.