Lesson 9
Polygons
Let’s investigate polygons and their areas.
Problem 1
Select all the polygons.
![Six figures labeled A, B, C, D, E, F.](https://cms-im.s3.amazonaws.com/zwbWtpTZeaa4mFmd8mjcGr1v?response-content-disposition=inline%3B%20filename%3D%226-6.1.D.PP_Image_1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.D.PP_Image_1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233653Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=978b15656f328380f414c94d6e5a3cdb04f8644413436e32e4a357b81d97ec10)
A
B
C
D
E
F
Problem 2
Mark each vertex with a large dot. How many edges and vertices does this polygon have?
![12 sided polygon resembling a star](https://cms-im.s3.amazonaws.com/LD3y7y9YwqxkdKrPQXJWQXff?response-content-disposition=inline%3B%20filename%3D%226-6.1.D.PP_Image_2.1.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.D.PP_Image_2.1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233653Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6e8ae2bcb336aa0dc384076a6a68683ea4af2249a3363683e4fb3c8eab9e5ea1)
Problem 3
Find the area of this trapezoid. Explain or show your strategy.
![Trapezoid, bases 8 and 4 units. Height 3 units.](https://cms-im.s3.amazonaws.com/avt3EEccdUpiJ2xz7Gduju2X?response-content-disposition=inline%3B%20filename%3D%226-6.1.D.PP_Image_4.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.D.PP_Image_4.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233653Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=726037b7142fedcdfe03b4dd5d7f291d1e0e01ceca8cd10ce78675af1d034031)
Problem 4
Lin and Andre used different methods to find the area of a regular hexagon with 6-inch sides. Lin decomposed the hexagon into six identical, equilateral triangles. Andre decomposed the hexagon into a rectangle and two triangles.
![2 identical hexagons labeled Lin’s method and Andre’s method.](https://cms-im.s3.amazonaws.com/zS6MDGFZ6zDKUinTZDYPAZFE?response-content-disposition=inline%3B%20filename%3D%226-6.1.C.PP.New.Image.17-18.png%22%3B%20filename%2A%3DUTF-8%27%276-6.1.C.PP.New.Image.17-18.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233653Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fef62957cb9b61bf3b97aa0e9efd148a819756ecd757b40c4f89b9257c3e52ce)
Find the area of the hexagon using each person’s method. Show your reasoning.
Problem 5
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Identify a base and a corresponding height that can be used to find the area of this triangle. Label the base \(b\) and the corresponding height \(h\).
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Find the area of the triangle. Show your reasoning.
Problem 6
On the grid, draw three different triangles with an area of 8 square units. Label the base and height of each triangle.
![A blank coordinate plane with 16 evenly spaced horizontal units and 12 evenly spaced vertical units.](https://cms-im.s3.amazonaws.com/Mtje8WXsjQDE9fQ8VtAvNX1D?response-content-disposition=inline%3B%20filename%3D%226-6.7.C.PP.Image.00.Blank-Grid.png%22%3B%20filename%2A%3DUTF-8%27%276-6.7.C.PP.Image.00.Blank-Grid.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T233653Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6cb77cfd2329d7712374ffffc6b56bd98965984812b687df642ec9d5745a8acf)