The rectangle in the image has been scaled by a factor of 2. Each side of the perimeter is 2 times as long, so the perimeter has doubled, and the total area of the scaled rectangle can be made from 4 copies of the original rectangle.

Exploration

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Create a prism with a length, width, and height of 2 units. Then create a prism with a length, width, and height of 4 units. What is the scale factor to change the side lengths? How many times larger is the surface area of the larger cube? How about the volume?
Create a prism with a length, width, and height of 2 units. Then create a prism with a length, width, and height of 6 units. What is the scale factor to change the side lengths? How many times larger is the surface area of the larger cube? How about the volume?
Create a prism with a length, width, and height of 2 units. Then create a prism with a length, width, and height of 8 units. What is the scale factor to change the side lengths? How many times larger is the surface area of the larger cube? How about the volume?
What is the relationship between the scale factor to change the size of the cube, the surface area, and the volume?

Dilating a solid to produce a similar solid will scale each length dimension, affecting both the surface area and the volume of the solid.

A blue unit cube on the left of the diagram. A vertical arrow pointing upward and labeled times 2 is drawn along one of the vertical edges of the cube. A horizontal arrow pointing to the left labeled times 2 is drawn along one of the horizontal edges of the cube. A diagonal arrow pointing backwards and labeled times 2 is drawn along one of the diagonal edges of the cube. A larger cube 2 units cubes high, 2 unit cubes long, and 2 unit cubes high on the right of the diagram. The lower left unit cube is colored blue while the rest is colored green.

Scaling a solid by a linear scale factor of d causes each side length to scale by a factor of d. This means that

the surface area will scale by a factor of d^2
the volume will scale by a factor of d^3

The cube in the image has been scaled by a factor of 2. Each face now has 4 times the area of the initial faces, and the overall volume could fit 8 of the original cubes inside.

Keep in mind that scaling all of the length dimensions of a figure by a common scale factor results in a similar figure, but only scaling one dimension does not. Adding or subtracting the same number from all dimensions will also not create similar figures.

Examples

Example 1

Consider the square pyramid shown below:

A square based pyramid with base side label 'a', slant height label 's' and height label 'h'.

Dilate each dimension of the square pyramid by a scale factor of d. What happens to the total surface area of the figure?

Worked Solution

Create a strategy

Use the given dimensions to write an equation for the surface area of the square pyramid. Recall that the surface area of a pyramid is calculated with the equation SA = B + nA, where B is the area of the base polygon, n is the number of sides of the base polygon, and A is the area of one triangular face.

Then, dilate each dimension of the square pyramid by a scale factor of d, and write an equation to represent the surface area of the new solid.

Finally, compare the two surface area equations.

Apply the idea

For the surface area of the square pyramid, we have SA= a^{2}+ 4 \cdot \frac{1}{2} as = a^{2} + 2as

By dilating the dimensions of the pyramid, we know that each dimension will be multiplied by a scale factor of d. For the new surface area, we have SA_d = \left( ad \right)^{2} + 4 \cdot \frac{1}{2} ad \cdot sd = a^{2}d^{2} + 2 asd^{2} = d^{2} \left( a^{2} + 2as \right)

Comparing the equations shows us that the surface area of the dilated figure is d^2 times bigger than the original figure. When each dimension of the pyramid is dilated by a scale factor of d, its surface area is dilated by a scale factor of d^2.

Reflect and check

This ratio of area makes sense because we know that area of a 2D figure will increase by a scale factor of d^2 when the dimensions increase by a scale factor of d.

Dilate each dimension of the square pyramid by a scale factor of d. What happens to the total volume of the dilated figure?

Worked Solution

Create a strategy

Recall that the formula for calculating the volume of a pyramid is V= \dfrac{1}{3}Bh where B is the area of the base polygon and h is the height of the pyramid. Write the equation to represent the volume of the given pyramid.

Then, apply a scale factor of d to each dimension and write the equation to represent the volume again and compare the equations.

Apply the idea

For the volume of the square pyramid, we have V= \frac{1}{3} a^2h

By dilating the dimensions of the pyramid, we know that each dimension will be multiplied by a scale factor of d. For the new volume, we have V_d= \frac{1}{3}\left(ad \right)^{2} \cdot hd= \frac{1}{3} a^{2}d^{2}hd = \frac{1}{3}a^{2}hd^{3}= \left(\frac{1}{3}a^{2}h \right) d^{3}

When each dimension of the pyramid is dilated by a scale factor of d, its volume is dilated by a scale factor of d^3.

Example 2

Lakendra likes baking miniature cakes. First she makes a full size cake with volume 250 \text{ in}^3. Then she makes a miniature cake with volume 2 \text{ in}^3.

What is the scale factor?

Worked Solution

Create a strategy

To calculate the scale factor, we need to evaluate the cube root of the ratio of the miniature cake to the full size cake.

Apply the idea

\displaystyle \text{Scale factor}	\displaystyle =	\displaystyle \sqrt[3]{\left(\frac{2}{250}\right)}
\displaystyle {}	\displaystyle =	\displaystyle \sqrt[3]{\left(\frac{1}{125}\right)}	Simplify the fraction
\displaystyle {}	\displaystyle =	\displaystyle \frac{1}{5}	Evaluate the cube root

The miniature cake is \dfrac{1}{5} the size of the full-sized cake.

Example 3

Consider these two cylinders:

two cylinders. first with height of 30 ft and radius of 10ft, the second has a height of 24ft and radius of 8ft.

Are the cylinders similar? Explain.

Worked Solution

Create a strategy

To determine if the cylinders are similar, we will compare the ratios of corresponding dimensions (height and radius) between the two cylinders. If these ratios are equal, the cylinders are similar.

Apply the idea

For the first cylinder, the height is 30\text{ ft} and the radius is 10\text{ ft}. For the second cylinder, the height is 24\text{ ft} and the radius is 8 \text{ ft}. The ratio of the heights is 30:24\text{ or }5:4The ratio of the radii is 10:8\text{ or }5:4 Since both ratios are equal, the cylinders are similar.

Reflect and check

This comparison reveals a fundamental property of similarity in geometry: objects are similar if their corresponding dimensions are in proportion. In this case, the constant ratio of the dimensions indicates that one of the cylinders was dilated, thus creating a similar cylinder.

Find the ratio of the surface area of the smaller cylinder to the larger cylinder.

Worked Solution

Create a strategy

The surface area of a cylinder is given by 2\pi r\left(h+r\right), where r is the radius and h is the height. To find the ratio of the surface areas, we calculate the surface area for each cylinder, then form a ratio of the smaller to the larger.

Apply the idea

Surface area of the first cylinder: 2 \pi \left(10\right)\left(30+10\right)=800 \pi Surface area of the second cylinder:2 \pi \left(8\right)\left(24+8\right)=512\pi The ratio of the smaller to the larger surface area is 512\pi : 800 \pi = 16:25

Reflect and check

Comparing the height and radius from the smaller cylinder to the bigger cylinder, we found that they each have a ratio of 4:5. Squaring this ratio, we would get 4^{2}:5^{2} or 16:25, which is the ratio of the surface areas. This shows that if the ratios between dimensions of similar figures is a:b, the ratio of their areas is a^{2}:b^{2}.

Find the ratio of the volume of the smaller cylinder to the larger cylinder.

Worked Solution

Create a strategy

The volume of a cylinder is given by \pi r^{2}h. To find the ratio of the volumes, we calculate the volume for each cylinder and then form a ratio of the smaller to the larger.

Apply the idea

Volume of the first cylinder:\pi \left(10\right)^{2} \left(30\right)=3000\pi Volume of the second cylinder: \pi \left(8\right)^{2} \left(24\right)=1536\pi The ratio of the smaller to the larger volume is 1536\pi : 3000\pi = 64:125

Reflect and check

The cylinders' height and radius each had a ratio of 4:5. Cubing this ratio, we get 4^{3}:5^{3} or 64:125. This shows that if the ratio between the dimensions of similar figures is a:b, the ratio of their volumes is a^{3}:b^{3}.

If the measurements of the smaller cylinder are doubled, calculate the new ratios for the surface area and volume of the cylinders.

Worked Solution

Create a strategy

Doubling the measurements of the smaller cylinder affects its surface area and volume. The new dimensions will be used to calculate the updated surface area and volume, and then compare them to the original larger cylinder.

Apply the idea

After doubling, the smaller cylinder's height becomes 48\text{ ft} and the radius 16\text{ ft}.

New surface area: 2\pi\left(16\right)\left(48+16\right)=2048\pi

New volume: \pi \left(16\right)^{2} \left(48\right)=12\,288\pi

New surface area ratio: 2048\pi:800\pi=64:25

New volume ratio: 12\,288\pi:3000\pi=512:125

Reflect and check

Let's take a look at how the ratios of surface areas and volumes compare to the scale factor. We can use corresponding parts of the cylinders to determine if they are similar and find scale factor.

After doubling the smaller cylinder, the new radius is 16\text{ ft} and the new height is 48\text{ ft}
The original large cylinder has a radius of 10\text{ ft} and a height of 30\text{ ft}

The ratio of the heights is 48:30\text{ or }8:5

The ratio of the raii is 16:10\text{ or }8:5 So, the cylinders are similar with a ratio between the dimension of 8:5.

With similar figures, scaling the dimensions affects the area by the square of the scale factor.

8^{2}:5^{2}\text{ or }64:25 We can confirm this was our new surface area ratio.

Scaling the dimensions affects the volume by the cube of the scale factor.

8^3:5^3\text{ or }512:125 We can confirm this was our new surface area ratio.

Idea summary

For two-dimensional figures, the scale factor can be summarized in the following table:

Ratio (scale factor) of sides	Ratio of perimeters	Ratio of area
d	d	d^2

For three-dimensional figures, the scale factor can be summarized in the following table:

Ratio (scale factor) of sides	Ratio of surface areas	Ratio of volume
d	d^2	d^3

Outcomes

G.DF.2

The student will determine the effect of changing one or more dimensions of a three-dimensional geometric figure and describe the relationship between the original and changed figure.

G.DF.2d

Compare ratios between side lengths, perimeters, areas, and volumes of similar figures.

G.DF.2e

Recognize when two- and three-dimensional figures are similar and solve problems, including those in context, involving attributes of similar geometric figures.

10.06 Similarity in 2D and 3D figures

Ideas

Similarity in 2D and 3D figures

Exploration

Examples

Example 1

Example 2

Example 3

Outcomes

G.DF.2

G.DF.2d

G.DF.2e

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