Now, repeat the process for additional regular polygons. In the dynamic geometry sketch, you can also change the number of sides of the figure by clicking on "more" to add more sides to the regular polygon, or "less" to reduce the number of sides in the regular polygon.

Use the sketch to generate several different regular pentagons. Record the radius and apothem of 5 of these regular pentagons in a table like the one shown. You may create the table in your notes.

Once you have recorded the values for the radius and apothem, calculate the value of the cosine of 180 degrees divided by the number of sides 180° number of sides , where the number of sides is the number of sides in the regular polygon (in this case, a regular pentagon). Use that value to fill in the middle column of the table. How do the values in the middle column compare to the values of the apothem given to you by the sketch?

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Check Your Answer

The values in the middle column, radius × cos (180 degrees over number of sides 180° number of sides ) are equal to the length of the apothem.

Take Notes Activity

Use your notes to respond to the following questions:

1. How can you generalize your formula to find the apothem, a, of any regular polygon when you know the radius, r, and the number of sides, n?
   
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   Check Your Answer

   a = r cos( 180 over n 180 n )

   
2. Justify the formula using what you know about right triangle trigonometry.
   
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   Need a hint?

   Use a right triangle whose legs are the apothem and half of a side length and whose hypotenuse is the radius.

   How many of these triangles will fit into the central angle of the polygon?

   Think about cos (180 degrees over number of sides 180° number of sides ) as cos (360 degrees over 2 times number of sides 360° 2 × number of sides )