Mathematics Question and Solution
Let the side length of the regular pentagon be 4 units.
a = ⅕*180(5-2)
a = 108°
a is the single interior angle of the regular pentagon.
b = ½(180-a)
b = ½(180-108)
b = 36°
c = 2 units.
d = 90-36
d = 54°
tan54 = e/2
e = 2.7527638409 units.
Area yellow is;
0.5*2*2.7527638409
= 2.7527638409 square units.
Calculating area regular pentagon.
f² = 4²+4²-2*4*4cos108
f = 6.472135955 units.
g = 108-90
g = 18°
sin18 = h/4
h = 1.2360679775 units.
cos18 = j/4
j = 3.8042260652 units.
Therefore, area regular pentagon is;
(0.5*4*4sin108)+½(4+6.472135955)*3.8042260652
= 7.6084521304+19.9191862792
= 27.5276384096 square units.
The fraction of the regular pentagon that is shaded is;
2.7527638409÷27.5276384096
= 1/10
