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10.3: Areas of Similar Polygons

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Learning Objectives

  • Understand the relationship between the scale factor of similar polygons and their areas.
  • Apply scale factors to solve problems about areas of similar polygons.

Review Queue

  1. Are two squares similar? Are two rectangles?
  2. Find the scale factor of the sides of the similar shapes. Both figures are squares.
  3. Find the area of each square.
  4. Find the ratio of the smaller square’s area to the larger square’s area. Reduce it. How does it relate to the scale factor?

Know What? One use of scale factors and areas is scale drawings. This technique takes a small object, like the handprint to the right, divides it up into smaller squares and then blows up the individual squares. In this Know What? you are going to make a scale drawing of your own hand. Either trace your hand or stamp it on a piece of paper. Then, divide your hand into 9 squares, like the one to the right, probably 2 \ in \times 2 \ in. Take a larger piece of paper and blow up each square to be 6 \ in \times 6 \ in (meaning you need at least an 18 in square piece of paper). Once you have your 6 \ in \times 6 \ in squares drawn, use the proportions and area to draw in your enlarged handprint.

Areas of Similar Polygons

In Chapter 7, we learned about similar polygons. Polygons are similar when the corresponding angles are equal and the corresponding sides are in the same proportion. In that chapter we also discussed the relationship of the perimeters of similar polygons. Namely, the scale factor for the sides of two similar polygons is the same as the ratio of the perimeters.

Example 1: The two rectangles below are similar. Find the scale factor and the ratio of the perimeters.

Solution: The scale factor is \frac{16}{24}, which reduces to \frac{2}{3}. The perimeter of the smaller rectangle is 52 units. The perimeter of the larger rectangle is 78 units. The ratio of the perimeters is \frac{52}{78}=\frac{2}{3}.

The ratio of the perimeters is the same as the scale factor. In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor.

Example 2: Find the area of each rectangle from Example 1. Then, find the ratio of the areas.

Solution:

A_{small} &= 10 \cdot 16=160 \ units^2\\A_{large} &= 15 \cdot 24=360 \ units^2

The ratio of the areas would be \frac{160}{360}=\frac{4}{9}.

The ratio of the sides, or scale factor was \frac{2}{3} and the ratio of the areas is \frac{4}{9}. Notice that the ratio of the areas is the square of the scale factor. An easy way to remember this is to think about the units of area, which are always squared. Therefore, you would always square the scale factor to get the ratio of the areas.

Area of Similar Polygons Theorem: If the scale factor of the sides of two similar polygons is \frac{m}{n}, then the ratio of the areas would be \left( \frac{m}{n} \right)^2.

Example 2: Find the ratio of the areas of the rhombi below. The rhombi are similar.

Solution: There are two ways to approach this problem. One way would be to use the Pythagorean Theorem to find the length of the 3^{rd} side in the triangle and then apply the area formulas and make a ratio. The second, and easier way, would be to find the ratio of the sides and then square that. \left( \frac{3}{5} \right)^2=\frac{9}{25}

Example 3: Two trapezoids are similar. If the scale factor is \frac{3}{4} and the area of the smaller trapezoid is 81 \ cm^2, what is the area of the larger trapezoid?

Solution: First, the ratio of the areas would be \left( \frac{3}{4} \right)^2= \frac{9}{16}. Now, we need the area of the larger trapezoid. To find this, we would multiply the area of the smaller trapezoid by the scale factor. However, we would need to flip the scale factor over to be \frac{16}{9} because we want the larger area. This means we need to multiply by a scale factor that is larger than one. A=\frac{16}{9} \cdot 81=144 \ cm^2.

Example 4: Two triangles are similar. The ratio of the areas is \frac{25}{64}. What is the scale factor?

Solution: The scale factor is \sqrt{\frac{25}{64}}=\frac{5}{8}.

Example 5: Using the ratios from Example 3, find the length of the base of the smaller triangle if the length of the base of the larger triangle is 24 units.

Solution: All you would need to do is multiply the scale factor we found in Example 3 by 24.

b=\frac{5}{8} \cdot 24=15 \ units

Know What? Revisited You should end up with an 18 \ in \times 18 \ in drawing of your handprint.

Review Questions

Determine the ratio of the areas, given the ratio of the sides of a polygon.

  1. \frac{3}{5}
  2. \frac{1}{4}
  3. \frac{7}{2}
  4. \frac{6}{11}

Determine the ratio of the sides of a polygon, given the ratio of the areas.

  1. \frac{1}{36}
  2. \frac{4}{81}
  3. \frac{49}{9}
  4. \frac{25}{144}

This is an equilateral triangle made up of 4 congruent equilateral triangles.

  1. What is the ratio of the areas of the large triangle to one of the small triangles?
  2. What is the scale factor of large to small triangle?
  3. If the area of the large triangle is 20 \ units^2, what is the area of a small triangle?
  4. If the length of the altitude of a small triangle is 2 \sqrt{3}, find the perimeter of the large triangle.
  5. Carol drew two equilateral triangles. Each side of one triangle is 2.5 times as long as a side of the other triangle. The perimeter of the smaller triangle is 40 cm. What is the perimeter of the larger triangle?
  6. If the area of the smaller triangle is 75 \ cm^2, what is the area of the larger triangle from #13?
  7. Two rectangles are similar with a scale factor of \frac{4}{7}. If the area of the larger rectangle is 294 \ in^2, find the area of the smaller rectangle.
  8. Two triangles are similar with a scale factor of \frac{1}{3}. If the area of the smaller triangle is 22 \ ft^2, find the area of the larger triangle.
  9. The ratio of the areas of two similar squares is \frac{16}{81}. If the length of a side of the smaller square is 24 units, find the length of a side in the larger square.
  10. The ratio of the areas of two right triangles is \frac{2}{3}. If the length of the hypotenuse of the larger triangle is 48 units, find the length of the smaller triangle’s hypotenuse.

Questions 19-22 build off of each other. You may assume the problems are connected.

  1. Two similar rhombi have areas of 72 \ units^2 and 162 \ units^2. Find the ratio of the areas.
  2. Find the scale factor.
  3. The diagonals in these rhombi are congruent. Find the length of the diagonals and the sides.
  4. What type of rhombi are these quadrilaterals?
  5. The area of one square on a game board is exactly twice the area of another square. Each side of the larger square is 50 mm long. How long is each side of the smaller square?
  6. The distance from Charleston to Morgantown is 160 miles. The distance from Fairmont to Elkins is 75 miles. Charleston and Morgantown are 5 inches apart on a map. How far apart are Fairmont and Elkins on the same map.
  7. Marlee is making models of historic locomotives (train engines). She uses the same scale for all of her models. The S1 locomotive was 140 ft long. The model is 8.75 inches long. The 520 Class locomotive was 87 feet long. What is the scale of Marlee’s models? How long is the model of the 520 Class locomotive?
  8. Tommy is drawing a floor plan for his dream home. On his drawing, 1cm represents 2 ft of the actual home. The actual dimensions of the dream home are 55 ft by 40 ft. What will the dimensions of his floor plan be? Will his scale drawing fit on a standard 8.5 in by 11 in piece of paper? Justify your answer.
  9. Anne wants to purchase advertisement space in the school newspaper. Each square inch of advertisement space sells for $3.00. She wants to purchase a rectangular space with length and width in the ratio 3:2 and she has up to $50.00 to spend. What are the dimensions of the largest advertisement she can afford to purchase?
  10. Aaron wants to enlarge a family photo from a 5 by 7 print to a print with an area of 140 inches. What are the dimensions of this new photo?
  11. A popular pizza joint offers square pizzas: Baby Bella pizza with 10 inch sides, the Mama Mia pizza with 14 inch sides and the Big Daddy pizza with 18 inch sides. If the prices for these pizzas are $5.00, $9.00 and $15.00 respectively, find the price per square inch of each pizza. Which is the best deal?
  12. Krista has a rectangular garden with dimensions 2 ft by 3 ft. She uses \frac{2}{3} of a bottle of fertilizer to cover this area. Her friend, Hadleigh, has a garden with dimensions that are 1.5 times as long. How many bottles of fertilizer will she need?

Review Queue Answers

  1. Two squares are always similar. Two rectangles can be similar as long as the sides are in the same proportion.
  2. \frac{10}{25} = \frac{2}{5}
  3. A_{small} = 100, A_{large} = 625
  4. \frac{100}{625} = \frac{4}{25}, this is the square of the scale factor.

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