## Area of an Octagon

## Opening

# Area of an Octagon

How can you find the area of this octagon?

How can you find the area of this octagon?

Find the area of regular polygons.

All of these figures are regular polygons. In each figure, a vertical line has been drawn from the center of the figure to the midpoint of a side. This line is called the *apothem*. Note that the center of a regular polygon is the same distance from the midpoint of any side. Choose one of these figures, then:

- Use the length of the apothem and the side length to find the area of the polygon. (To get started, think about how you could divide the polygon into shapes that you know how to find the area of.)
- If you have time, find the area of a second polygon.

- How could you divide the polygon into triangles?
- How could you divide the polygon into parallelograms?
- How could you use the triangles to make a parallelogram that represents the total area of the polygon?

Look at this floor plan of a build-it-yourself log cabin. Assume the cabin is a regular polygon.

- What is the area of the cabin?

Prepare a presentation about finding the area of regular polygons. Support your thinking with your work.

If the area of a regular hexagon is 64.5 square inches, and the apothem is 4.3 inches, how long is each side?

Take notes about other students’ approaches to finding the area of regular polygons.

As your classmates present, ask questions such as:

- How is the apothem like the radius of a circle?
- How was your approach to finding the area of the pentagon different from your approach to finding the areas of the other polygons? How was it the same?
- What did you do first when solving the cabin problem—that is, how did you make sense of the problem?

Write a summary about finding the area of regular polygons.

Check your summary.

- Do you explain how to find the area of a regular polygon in at least two ways?
- Do you explain how a regular polygon is different from a polygon that is not regular, in a way that makes it easier to find its area?

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Some places I see regular polygons are …**