New York Geometry - 2020 Edition
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6.05 Perimeters and areas of similar figures
Lesson

Recall that two figures are similar if corresponding angles are congruent and corresponding sides are proportional. All pairs of corresponding sides in similar figures are in the same ratio, and we can use this ratio to find the scale factor.

Since this scale factor affects the length of sides, it is also called the length scale factor or linear scale factor.

But what happens to the area of a figure when we enlarge it by a linear scale factor? Does it also enlarge? Is it by the same or some other related scale factor?

Suppose we have a square with side lengths $2$2 cm, shown in red below, and we enlarge it by a length scale factor of $3$3, shown in blue below.

Our lengths have been scaled by a factor $3$3, but our area has gone from $4$4 cm2 to $36$36 cm2. It has been scaled by a factor of $9$9!

Areas of similar figures do not scale by the same factor as the linear scale factor, and for this reason, they have their own scale factor called the area scale factor.

So, is there any way to predict what the area scale factor of a shape will be if we know its length scale factor? The answer is yes, and we'll see why.

 

Exploration

Let's say we have a square of side length $a$a units. This square would therefore have area $a^2$a2 units2.

Let's say we want to scale it by a length scale factor of $k>1$k>1 (this value can actually be less than one, in which case the square would shrink, but let's just look at $k>1$k>1 for now).

What is the area now?

The side length of our new square is $a\times k$a×k units. We can use this to figure out the new area.

New Area $=$= $\left(a\times k\right)\times\left(a\times k\right)$(a×k)×(a×k) units2
  $=$= $\left(a\times k\right)^2$(a×k)2 units2
  $=$= $a^2\times k^2$a2×k2 units2

So, if our old area was $a^2$a2 units2 and our new area is $a^2\times k^2$a2×k2 units2, we have scaled our area by a factor of $k^2$k2. Since this works for any side length $a$a units and any scale factor $k$k, we know that if we scale any square by length scale factor $k$k, the area scale factor will always be $k^2$k2.

 

Area in similar figures

For any figure that is scaled by a linear scale factor of $k$k, the area will scale by an area scale factor of $k^2$k2.

 

Practice questions

Question 1

Find the value of $x$x in each of the following cases:

Question 2

Consider the two similar rectangles shown in the figure.

  1. Find the area of Rectangle A.

    Area A = $\editable{}$cm2

  2. Find the area of Rectangle B.

    Area B = $\editable{}$cm2

  3. What is the ratio of the area of Rectangle A to Rectangle B?

  4. Is it true in this case that if the matching sides of two similar figures are in the ratio $m:n$m:n, then their areas are in the ratio $m^2:n^2$m2:n2?

    True

    A

    False

    B

    True

    A

    False

    B

 

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