# Lesson 1Planning the GazeboDevelop Understanding

## Jump Start

An equilateral triangle can be folded in half to form two congruent triangles.

Find the area of the equilateral triangle in two different ways.

### 1.

Using right triangle trigonometry:

### 2.

Using the Pythagorean theorem:

## Learning Focus

Find the perimeter and area of regular polygons.

How can we extend our list of area formulas to include the set of all regular polygons?

Do we need a different rule for each type of regular polygon, or can we find a general method for finding the area of any regular polygon?

## Open Up the Math: Launch, Explore, Discuss

Zac is using his knowledge of geometry to design a gazebo for his family’s backyard. The gazebo will be in the shape of a regular polygon. As part of his design, Zac will need to calculate several things so that his parents can purchase the right amount of wood for the construction. Zac will need to calculate:

• the perimeter of the gazebo so he can order enough railing to surround it.

• the area of the floor of the gazebo so he can order enough planks to lay it.

• the surface area of the pyramid, which forms the roof that will cover it.

The problem is, his parents keep changing their minds about what shape they would like the gazebo to be—a hexagon, an octagon, a decagon, a dodecagon, or even some other type of -gon.

From his work with symmetries of polygons, Zac knows that all regular polygons are cyclic—that is, every regular polygon can be inscribed in a circle. Zac is wondering if he can use this property of regular polygons to help him find their perimeter and area.

For his first attempt at creating a scale drawing of the gazebo, Zac has inscribed a regular hexagon inside a circle with a radius of inches. He is not sure if this is enough information to find the perimeter of this hexagon and the area it encloses.

### 1.

To start the task of finding the perimeter of this hexagon, Zac decides to write down what he thinks he knows. Decide if you agree or disagree with each of his statements, and explain why. You will also need to add features to the diagram to illustrate Zac’s comments.

What Zac thinks he knows:

Do you agree or disagree? Explain why.

Two radii drawn to two consecutive vertices of the regular hexagon form a central angle whose measure can be found based on the rotational symmetry of the figure.

The hexagon can be decomposed into 6 congruent isosceles triangles.

The length of the altitudes of each of these 6 congruent triangles (the altitude drawn from the vertex of the triangle, which is located at the center of the circle) can be found using trigonometry.

The length of the sides of the triangle that form chords of the circle can be found using trigonometry.

### 2.

Based on what you and Zac know, find the perimeter of the hexagon that he inscribed in the circle with a radius of inches. Describe your strategy so someone else can follow it.

### 3.

Now find the area of the hexagon that Zac inscribed in the circle with a radius of inches. Describe your strategy so someone else can follow it.

### 4.

What if Zac had inscribed an octagon inside the circle of radius instead of a hexagon? Modify your strategy to find the perimeter and area of the octagon.

### 5.

Modify your strategy to find the perimeter and area of any regular -gon inscribed in a circle of any given radius.

### 1.

Use your formula for the perimeter of a regular polygon to find the perimeter of each of the following polygons inscribed in a circle with a radius of :

#### a.

A regular pentagon

#### b.

A regular decagon

A regular

A regular

### 2.

What did you notice?

## Takeaways

I can use a strategy or a formula to find the perimeter or area of a regular polygon.

My description of the strategy:

The formulas that generalize this strategy:

• For perimeter of a regular polygon:

• For area of a regular polygon:

## Vocabulary

• n-gon
• Bold terms are new in this lesson.

## Lesson Summary

In this lesson, we developed a strategy for finding the perimeter and area of a regular polygon. We first inscribed the regular polygon in a circle so we could draw upon our knowledge of central angles, radii, and chords. By decomposing the regular polygon into right triangles, we could use trigonometric ratios to find the lengths that we needed to calculate perimeter and area.

## Retrieval

### 1.

Given the area of a circle, find the measure of the radius and the circumference.

 radius $=$ area $=$ circumference $=$

### 2.

Find the perimeter and area of the figure.