**Content Objective:**

The following problems can be used at the end of the lesson to introduce realistic applications for area and perimeter of regular polygons. Additional questions to each problem could be included to extend each of the problems as necessary.

**Problems:**

**Standard Soccer Ball**

A standard soccer ball has 60 vertices, 20 regular hexagons and 12 regular pentagons, all with 90 equal edges.

Allow students to measure the side length of regular soccer ball if available, or give students side length.

Questions:

1. What is the area of ONE of the regular pentagons?

2. What is the area of ONE of the regular hexagons?

3. What is the surface area of the ENTIRE soccer ball?

Click here for a larger image of the standard soccer ball.

**Tiling Pattern Variation**

Our new tiling pattern has 12 regular 10-sided polygons, 22 regular hexagons and 22 squares, all with equal edges.

Give students a side length.

Questions:

1. What is the area of ONE of the regular 10-sided polygons?

2. What is the area of ONE of the regular hexagons? One square?

3. What is the surface area of the ENTIRE soccer ball with the new tiling pattern?

Click here for a larger image of the standard soccer ball.

For more information and sphere tiling patterns, please visit the following website: Higher Genus "Soccer Balls" Picture Page.

**Concert Hall**

The local concert hall committee is planning to remodel the stage and storage area of the concert hall. Before they can begin to redesign these spaces, they must provide the local architect with certain measurements of the entire building. The concert hall committee knows the wall shown below is 50ft, and the entire floor plan is composed of regular polygons.

Use the diagram below to calculate the following measurements. It may be helpful to divide the larger spaces in the concert hall into small regular polygons.

Questions:

1. What is the area of the stage and the storage room?

2. What is the area of the remaining spaces?

3. What is the area of the entire concert hall? What percentage of area will be remodeled?

4. How many exterior walls does the concert hall have? Find the perimeter of the entire concert hall.

Click here to see the GSP file for the concert hall.

**Office Building**

The local art museum has relocated and left behind the strange building shown in the diagram below. The new owners would like to turn this space into an office building. Before drawings can be sent to the architect for redesign, the building measurements and dimensions must be determined. Assume this building can be broken into regular polygons, and that one of the walls has a length of 50 feet.

Use the diagram below to calculate the following measurements. First divide the office building into smaller regular polygons. Label your different spaces of the office building (i.e. label the entrance, restrooms, offices, kitchen, etc.). Be creative!

Questions:

1. How many regular polygons make up the office building floor plan? How many triangles, squares, pentagons, etc.?

2. What is the area of the entire office building?

3. What is the perimeter of the entire office building?

4. How many ways can you divide this floor plan into regular polygons? Is there more than one way? Explain your answer.

5. How would the area of the office building change if each wall length were larger? Smaller?

6. Assume we know the perimeter, but not the length of one wall. How many walls make up the perimeter of the office building? If the perimeter is 575 feet, what is the length of one wall?

Click here to see the GSP file for this problem.

For this connection activity, students will design their own floor plan for their home, made up of regular polygons meeting the following constraints:

-The perimeter of the entire floor plan should be between 180 feet and 220 feet.

Note- When building use the scale: 2 inches = 10 feet.-Side lengths should be 10 feet.

-Have students creatively label each room they construct!

Use cardboard, poster board, paper, ... for construction. Turn in floor plan along with a report including answers to the following:

1. How did you design your floor plan; what was your thought process?

2. What is the area of each room?

3. What is the area of the entire floor plan?

4. What is the perimeter of the floor plan?

Allow students to present their floor plans to the class. This will allow students to see that their are many possibilities for this assignment.

Click here to see a sample floor plan on GSP. (However, remember that student floor plans should not be drawn, but rather constructed out of material.)

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