10.12. Area of Regular Polygons http://www.ck12.org
To find the length of the apothem,AB, you will need to use the trig ratios. First, findm^6 CAD. There are 360◦around
a point, som^6 CAD=^360
◦
8 =^45
◦. Now, we can use this to find the other two angles in 4 CAD.m (^6) ACBandm (^6) ADC
are equal because 4 CADis a right triangle.
m^6 CAD+m^6 ACB+m^6 ADC= 180 ◦
45 ◦+ 2 m^6 ACB= 180 ◦
2 m^6 ACB= 135 ◦
m^6 ACB= 67. 5 ◦
To findAB, we must use the tangent ratio. You can use either acute angle.
tan 67. 5 ◦=
AB
6
AB= 6 ·tan 67. 5 ◦≈ 14. 49Watch this video for help with the Examples above.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/52483CK-12 Foundation: Chapter10AreaofRegularPolygonsB
Concept Problem Revisited
From the picture below, we can see that the total distance across the Pentagon is the length of the apothem plus
the length of the radius. If the total area of the Pentagon is 34 acres, that is 2,720,000 square feet. Therefore, the
area equation is 2720000=^12 a( 921 )( 5 )and the apothem is 590.66 ft. To find the radius, we can either use the
Pythagorean Theorem, with the apothem and half the length of a side or the sine ratio. Recall from Example 5, that
each central angle in a pentagon is 72◦, so we would use half of that for the right triangle.