Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 1: Transformations

Lesson 6: Perimeters and Areas of Similar Figures

In this Grade 7 lesson from Big Ideas Math, Course 2 Accelerated (Chapter 11), students explore how the perimeters and areas of similar figures change when dimensions are scaled by a constant factor. Through hands-on activities, students discover that when side lengths are multiplied by a scale factor, the perimeter changes by the same factor while the area changes by the square of that factor. Students also practice finding and comparing ratios of perimeters and areas of similar figures on a coordinate grid.

Section 1

Perimeter Ratio of Similar Figures

Property

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

\frac{\text{Perimeter of Figure 1}}{\text{Perimeter of Figure 2}} = \frac{\text{Side length of Figure 1}}{\text{Corresponding side length of Figure 2}}

Section 2

Areas of Similar Figures

Property

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

The new figure is similar to the original figure, and
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.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Perimeter Ratio of Similar Figures

Property

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

\frac{\text{Perimeter of Figure 1}}{\text{Perimeter of Figure 2}} = \frac{\text{Side length of Figure 1}}{\text{Corresponding side length of Figure 2}}

Section 2

Areas of Similar Figures

Property

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

The new figure is similar to the original figure, and
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.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Perimeter Ratio of Similar Figures

Property

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

\frac{\text{Perimeter of Figure 1}}{\text{Perimeter of Figure 2}} = \frac{\text{Side length of Figure 1}}{\text{Corresponding side length of Figure 2}}

Section 2

Areas of Similar Figures

Property

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

The new figure is similar to the original figure, and
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.