Take note: Postulate 10-1

If two figures are congruent, then their areas are equal.

Take note Theorem 10-6: Area of a Regular Polygon

The area of a regular polygon is half the product of the apothem and the perimeter.

A = 1 2 a p eh equals . 1 half , eh p

Problem 2 Finding the Area of a Regular Polygon

What is the area of the regular decagon below?

• Step 1 Find the perimeter of the regular decagon.

  p = n s Use the formula for the perimeter of an  n -gon .   = 10 ( 8 ) Substitute  10  for  n  and  8  for  s .   = 80  in .   table with 3 rows and 3 columns , row1 column 1 , p , column 2 equals n s , column 3 cap usetheformulafortheperimeterofan . n , minusgon , . , row2 column 1 , , column 2 equals 10 , open 8 close , column 3 cap substitute . 10 , for , n , and , 8 , for , s . , row3 column 1 , , column 2 equals 80 , in . , column 3 , end table

• Step 2 Find the area of the regular decagon.

  A = 1 2 a p Use the formula for the area of a regular polygon .   = 1 2 ( 12.3 ) ( 80 ) Substitute  12.3  for  a  and  80  for  p .   = 492   table with 3 rows and 3 columns , row1 column 1 , eh , column 2 equals , 1 half , eh p , column 3 cap usetheformulafortheareaofaregularpolygon . . , row2 column 1 , , column 2 equals , 1 half . open , 12.3 , close . open 80 close , column 3 cap substitute . 12.3 , for , eh , and , 80 , for , p . , row3 column 1 , , column 2 equals 492 , column 3 , end table

  The regular decagon has an area of 492   in . 2 . 492 in , . squared , .

✓ Got It?

2. 1. What is the area of a regular pentagon with an 8-cm apothem and 11.6-cm sides?
   2. Reasoning If the side of a regular polygon is reduced to half its length, how does the perimeter of the polygon change? Explain.