Topics
11. Modeling in Two & Three Dimensions
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United States of AmericaFL
Grade 912

11.07 Effects of changing dimensions

Lesson
Practice
Lesson

Concept summary

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.

Similarly, 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.

Worked examples

Example 1

A photo with dimensions 4 \text{ in} \times 6 \text{ in} is enlarged to dimensions 10 \text{ in} \times 15 \text{ in}.

a

Find the scale factor.

Approach

We look at the change in side lengths, using the formula;

\dfrac{\text{Enlarged photo side}}{\text{Original photo side}}

Solution

Finding the scale factors betweeen the short sides:

\displaystyle \text{Scale factor}\displaystyle =\displaystyle \dfrac{\text{Enlarged photo short side}}{\text{Original photo short side}}Formula
\displaystyle {}\displaystyle =\displaystyle \dfrac{10}{4}Substitution
\displaystyle {}\displaystyle =\displaystyle \dfrac{5}{2}Simplify

Finding the scale factors betweeen the longer sides:

\displaystyle \text{Scale factor}\displaystyle =\displaystyle \dfrac{\text{Enlarged photo long side}}{\text{Original photo long side}}Formula
\displaystyle {}\displaystyle =\displaystyle \dfrac{15}{6}Substitution
\displaystyle {}\displaystyle =\displaystyle \dfrac{5}{2}Simplify

Since the two scale factors we found above are equal, we say that the scale factor between the enlarged photo and original photo is \dfrac{5}{2}.

Reflection

To be able to say the shape has a scale factor, all sides must increase by the same scale factor.

b

How many times greater is the perimeter of the enlarged photo than the original image.

Approach

We found the scale factor in part (a), so we can use this to figure our how many times greater the perimeter of the enlarged photo is than the original photo.

If the lengths are scaled by a scale factor of d, then the perimeter will scale by a scale factor of d.

Solution

The perimeter of the enlarged photo will be \dfrac{5}{2} times larger than the perimeter of the original photo.

c

Describe the relationship between the areas of the two photos.

Approach

If the lengths are scaled by a scale factor of d, then the area will scale by a scale factor of d^2.

We want to use the scale factor from part (a) to find the scale factor between the areas.

Solution

The area scale factor is the linear scale factor squared, so the area scale factor is \left( \dfrac{5}{2} \right)^2 = \dfrac{25}{4}.

The enlarged photo has an area \dfrac{25}{4} times greater than the original photo.

Example 2

Lauren likes making 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.

Find the scale factor.

Approach

To work out the scale factor we need to take the cube root of the ratio.

Solution

\displaystyle \text{Scale factor}\displaystyle =\displaystyle \sqrt[3]{\left(\dfrac{2}{250}\right)}
\displaystyle {}\displaystyle =\displaystyle \sqrt[3]{\left(\dfrac{1}{125}\right)}Simplify
\displaystyle {}\displaystyle =\displaystyle \dfrac{1}{5}Simplify

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

Outcomes

MA.912.GR.1.6

Solve mathematical and real-world problems involving congruence or similarity in two-dimensional figures.

MA.912.GR.4.3

Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.

MA.912.GR.4.4

Solve mathematical and real-world problems involving the area of two-dimensional figures.

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