Concept Byte: Exploring Area and Circumference

Use With Lesson 10-7

TECHNOLOGY

A polygon is inscribed in a circle when all its vertices are on the circle. Work in small groups to investigate inscribed regular polygons. Record your results in a table like the one below.

Construct

• Step 1 Use geometry software to construct a circle. Find its circumference and area. Record them in your table.
• Step 2 Inscribe an equilateral triangle in the circle. Your software may do this automatically. If not, select three points that are approximately evenly spaced around the circle. Draw a triangle with these points as vertices.

  

Investigate

• Step 3 Use your geometry software to find the perimeter and area of the triangle. Record them in your table. Calculate and record the ratios

  triangle perimeter circle circumference  and  triangle area circle area . fraction triangle perimeter , over circle circumference end fraction . and . fraction triangle area , over circle area end fraction . .

• Step 4 Change the size of the circle. Do the ratios you calculate stay the same or do they change?
• Step 5 Repeat Steps 3 and 4 for a square and a regular pentagon.

Exercises

1. Make a Conjecture What will happen to the ratios as you increase the number of sides of the polygon? As you increase the size of the circle?

Regular Octagon

Extend

2. Extend your table to include a regular 12-sided polygon.
   1. Do your conjectures hold for the 12-sided polygon?
   2. Compare the ratios perimeter circumference  and  polygon area circle area fraction perimeter , over circumference end fraction . and . fraction polygon area , over circle area end fraction  in your table. How do they differ?
3. Estimation Estimate the perimeter and area of a polygon of 100 sides that is inscribed in a circle with a radius of 10 cm. Explain how you made your estimate.

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