11.2: From the Center of the Polygon
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.
Topic: Quadrilaterals & General Polygons
- Regular Polygons
- Area of Regular Polygons
Teacher Preparation and Notes
- This activity was written to be explored with the Cabri Jr. app on the TI-84.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11644 and select PENTAGON, HEXAGON, OCTAGON.
Associated Materials
- Student Worksheet: From the Center of a Polygon http://www.ck12.org/flexr/chapter/9695, scroll down to the second activity.
- Cabri Jr. Application
- PENTAGON.8xv, HEXAGON.8xv, OCTAGON.8xv
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 \begin{align*}ABCDE\end{align*} with center \begin{align*}R\end{align*}. Students are given the length of \begin{align*}\overline{CD}\end{align*} (side of the polygon), \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, \begin{align*}a \cdot p\end{align*} (apothem times perimeter), and Area.
Students will collect data by moving point \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
1. Sample answers:
Position | Apothem \begin{align*}(a)\end{align*} | Perimeter \begin{align*}(p)\end{align*} | \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 \begin{align*}= \frac{1}{2}\end{align*} (apothem)(perimeter) \begin{align*}= \frac{1}{2} ap\end{align*}
3. Sample answers:
Position | Apothem \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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