11.7: Extension: Exploring Similar Solids
Learning Objectives
- Find the relationship between similar solids and their surface areas and 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 they are the same type of solid and their corresponding radii, heights, base lengths, widths, etc. are proportional.
Example 1: Are the two rectangular prisms similar? How do you know?
Solution: Match up the corresponding heights, widths, and lengths.
\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{3}{4}=\frac{12}{16}\end{align*} however, \begin{align*}\frac{8}{12}=\frac{2}{3}\end{align*}.
These triangle pyramids are not similar.
Surface Areas of Similar Solids
If two shapes are similar, then the ratio of the area is a square of the scale factor.
For example, the two rectangles 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*}. The area of the smaller rectangle is \begin{align*}5(10)=50 \ units^2\end{align*}.
Comparing 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?
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\\ 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 are in a ratio of \begin{align*}\frac{2^2}{3^2}\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*}.
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*}. 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 area of the smaller cylinder is \begin{align*}1536 \pi \ cm^2\end{align*}, what is the 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. |
If the ratio of the volumes follows the pattern from above, it should be the cube of the scale factor.
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*}.
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. \begin{align*}3^3:4^3 = 27:64\end{align*}.
Example 8: If the ratio of the volumes of two similar prisms is 125:8, what is the scale factor?
Solution: 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 triangles.
Solution: The scale factor is 7:5, the cubed root. 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 10: The ratio of the surface areas of two similar cylinders is 16:81. What is the ratio of the volumes?
Solution: First, find the scale factor. If we take the square root of both numbers, the ratio is 4:9. Now, cube this to find the ratio of the volumes, \begin{align*}4^3 : 9^3 = 64:729\end{align*}.
Review Questions
- Questions 1-4 are similar to Examples 1 and 2.
- Questions 5-14 are similar to Examples 3-8 and 10.
- Questions 15-18 are similar to Example 9.
- Questions 19 and 20 are similar to Example 1.
Determine if each pair of right solids are similar.
- 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?
- 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. 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.
- 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 the volume of both pyramids.
Use the hemispheres below to answer questions 19-20.
- Are the two hemispheres similar? How do you know?
- Find the ratio of the surface areas and volumes.