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# 11.2: From the Center of the Polygon

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 10, Lesson 6.

ID: 11644

Time Required: 45 minutes

## Activity Overview

Students will explore the area of a regular polygon in terms of the apothem and the perimeter. They will derive the formula for a regular pentagon and regular hexagon. Then, students will see how the formula relates to the formula for the area of triangles. Students will then be asked to apply what they have learned about the area of a regular polygon.

• Regular Polygons
• Area of Regular Polygons

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Area of a Regular Pentagon

Students will begin this activity by looking at a regular pentagon. In file PENTAGON, students are given regular pentagon ABCDE\begin{align*}ABCDE\end{align*} with center R\begin{align*}R\end{align*}. Students are given the length of CD¯¯¯¯¯¯¯¯\begin{align*}\overline{CD}\end{align*} (side of the polygon), RM¯¯¯¯¯¯¯¯¯\begin{align*}\overline{RM}\end{align*} (apothem), and the area of the polygon. Students are to collect data in the table given on their student worksheet and contains 4 columns: Apothem, Perimeter, ap\begin{align*}a \cdot p\end{align*} (apothem times perimeter), and Area.

Students will collect data by moving point D\begin{align*}D\end{align*}. They will do this for four different positions of the point.

Students are asked about how the area and the apothem times the perimeter are related.

## Problem 2 – Area of a Regular Hexagon

Students will repeat the same process as Problem 1 for a regular hexagon. Students will begin to discover the formula for a regular polygon is one-half the perimeter times the apothem.

## Problem 3 – Area of a Regular Polygon

In this problem, students are to “prove” the formula for the area of a regular polygon by looking at an octagon and the triangles created by the radii of the octagon. If students are confused by the term radius of the polygon, explain that this is the radius of a circle circumscribed about a regular polygon.

## Problem 4 – Area of Regular Polygons

In Problem 4, students are asked to apply what they have learned about the area of regular polygons. The students are given a question on the area of a regular polygon and a calculator on each page. The students are to use the calculator to find the area.

## Solutions

Position Apothem (a)\begin{align*}(a)\end{align*} Perimeter (p)\begin{align*}(p)\end{align*} ap\begin{align*}a \cdot p\end{align*} (apothem times perimeter) Area
1 1.72 12.5 21.5 10.75
2 1.93 14 27.02 13.49
3 2.13 15.5 33.015 16.53
4 2.55 18.5 47.175 23.55

2. area =12\begin{align*}= \frac{1}{2}\end{align*} (apothem)(perimeter) =12ap\begin{align*}= \frac{1}{2} ap\end{align*}

Position Apothem (a)\begin{align*}(a)\end{align*} Perimeter \begin{align*}(p)\end{align*} \begin{align*}a \cdot p\end{align*} (apothem times perimeter) Area
1 1.91 13.2 25.2 12.6
2 2.25 15.6 35.1 17.76
3 2.34 16.2 37.908 18.9
4 2.60 18 46.8 23.4

4. Area \begin{align*}= \frac{1}{2}\end{align*} (apothem)(perimeter) \begin{align*}= \frac{1}{2} ap\end{align*}

5. 10

6. Yes

7. Yes

8. Area CDR \begin{align*}= \frac{1}{2} as\end{align*}

9. Area \begin{align*}= \frac{1}{2} as(8) = 4as\end{align*}

10. Area \begin{align*}= \frac{1}{2} asn\end{align*}

11. 485.52 sq. in.

12. 25.2 sq. cm

13. 501.84 sq. ft

14. 1,039.2 sq. mm

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