Mathematics Question and Solution
a = ⅕*180(5-2)
a = 108°
a is the single interior angle of the regular pentagon.
b = 180-a-42
b = 180-150
b = 30°
(12/sin108) = (c/sin30)
c = 6.30877334543 cm.
c is the side length of the regular pentagon.
d² = 2(6.30877334543)²-2(6.30877334543)²cos108
d = 10.2078097002 cm.
e = 108-½(180-108)
e = 108-36
e = 72°
Therefore, area regular pentagon is;
2(0.5*6.30877334543²cos(108))+(0.5*6.30877334543*10.2078097002sin(72)
= 37.8526400726+30.6234291006
= 68.4760691732 cm²
Calculating the required angle.
Let it be x.
(12/sin108) = (f/sin42)
f = 8.44278666802 cm.
g = f-c
g = 8.44278666802-6.30877334543
g = 2.13401332259 cm.
g is the length of the two equal short lengths.
h² = 2.13401332259²+10.2078097002²-2*2.13401332259*10.2078097002cos72
h = 9.76167943379 cm.
(9.76167943379/sin72) = 2.13401332259/sink
j = 12°
Therefore x, the required angle is;
x = 108-j-36
x = 108-12-36
x = 108-48
x = 60°
It implies;
Area regular pentagon is;
68.4760691732 cm²
Required angle is;
60°
