Lesson 6: Perimeters and Areas of Similar Figures

Property

To officially prove two polygons are similar without using transformations, you must test the ratios of ALL their corresponding sides. You create a fraction for each pair of sides: $\frac{\text{Image Side}}{\text{Pre-image Side}}$. If every single fraction simplifies to the exact same number (the scale factor $k$), the figures are similar.

Examples

• Testing Triangles: Triangle 1 has sides $3, 4, 5$. Triangle 2 has sides $6, 8, 10$.
  • Check ratios: $\frac{6}{3} = 2$, $\frac{8}{4} = 2$, $\frac{10}{5} = 2$. All equal $2$. They are similar!
• Testing Rectangles: Rectangle A is $4$ by $6$. Rectangle B is $6$ by $8$.
  • Check ratios: $\frac{6}{4} = 1.5$, but $\frac{8}{6} \approx 1.33$. The ratios are NOT equal. They are not similar.

Explanation

This is the ultimate test for similarity. A common mistake students make is only checking one pair of sides and assuming the whole shape is similar. You must check every pair! When setting up your fractions, be organized: always put the sides of the second shape on the top (numerator), and the sides of the first shape on the bottom (denominator). Make sure you are matching the shortest side to the shortest side, and the longest to the longest!

Property

For similar figures, the ratio of their perimeters equals the ratio of any pair of corresponding side lengths:

Property

If we multiply each dimension of a figure by $k$, then:

1. The new figure is similar to the original figure, and
2. The area of the new figure is $k^2$ times the area of the original figure.

Examples

• A square with a side length of 5 cm has an area of 25 cm$^2$. If you scale its dimensions by a factor of $k=3$, the new side is 15 cm and the new area is $15^2 = 225$ cm$^2$, which is $3^2 \times 25 = 9 \times 25$.

• A circular rug has a radius of 2 feet. A larger, similar rug has a radius of 6 feet. The scale factor is 3, so the area of the larger rug is $3^2=9$ times the area of the smaller one.