Posted by hope on Tuesday, April 10, 2012 at 12:07am.
Area of a regular polygon:
A = ( 1 / 2 ) * n * r ^ 2 * sin ( 360 ° / n )
The area of a decagon:
( n = 10 )
( r = 4 mm )
( sin 36 ° = sqrt [ 10 - 2 sqrt ( 5 ) ] / 4
A = ( 1 / 2 ) * 10 * 4 ^ 2 * sin ( 360 ° / 10 )
A = 10 / 2 * 16 * sin ( 36 ° )
A = 5 * 16 * sin ( 36 ° )
A = 5 * 16 * sqrt [ 10 - 2 sqrt ( 5 ) ] / 4
A = 5 * 16 / 4 * sqrt [ 10 - 2 sqrt ( 5 ) ]
A = 5 * 4 * sqrt [ 10 - 2 sqrt ( 5 ) ]
A = 20 * 2.351141
A = 47.02282 mm ^ 2
The area of a octagon:
( n = 8 )
( r = 10 mm )
( sin 45 ° = 1 / sqrt ( 2 ) = sqrt ( 2 ) / 2 )
A = ( 1 / 2 ) * 8 * 10 ^ 2 * sin ( 360 ° / 8 )
A = 8/ 2 * 100 * sin ( 45 ° )
A = 4 * 100 * sin ( 45 ° )
A = 400 * sqrt ( 2 ) / 2
A = ( 400 / 2 ) * sqrt ( 2 )
A = 200 * sqrt ( 2 )
A = 200 * 1.41421356
A = 282.842712 mm ^ 2
Find the area of the shaded portion of the diagram (use 3.14 for p).
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