Topics
9. Modeling in Two and Three dimensions
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United States of America
Grades 9-12

9.07 Similarity in 2D and 3D figures

Lesson
Practice

Introduction

We saw scale drawings in 7th grade, where we were introduced to calculating dimensions to make copies of scale drawings. That concept is extended here to using similarity in 2D figures and applying our understanding to 3D figures and solving problems.

Similarity in 2D and 3D figures

Recall that dilating a shape by a constant factor will scale the side lengths, affecting both the perimeter and the area of the shape.

A blue 2 by 3 rectangular array on the left of the diagram. A vertical arrow pointing to the right, and a horizontal arrow pointing down are drawn from the top left corner of the array, and both labeled times 2. A 4 by 6 rectangular array on the right of the diagram. The upper leftmost 2 by 3 array part is colored blue while the rest is colored green.

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

  • the perimeter will scale by a factor of d
  • the area will scale by a factor of d^2

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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  1. 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?

  2. 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?

  3. 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?

  4. 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.

Examples

Example 1

Consider the square pyramid shown below:

a

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.

b

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= \frac{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^2d^2hd = \frac{1}{3}a^2hd^3= \left(\frac{1}{3}a^2h \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(\dfrac{2}{250}\right)}
\displaystyle {}\displaystyle =\displaystyle \sqrt[3]{\left(\dfrac{1}{125}\right)}Simplify the fraction
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{5}Evaluate the cube root

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

Idea summary

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

Ratio (scale factor) of sidesRatio of perimetersRatio of area
ddd^2

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

Ratio (scale factor) of sidesRatio of surface areasRatio of volume
dd^2d^3

Outcomes

G.GMD.A.3

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

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