Mathematics Question and Solution
a = (180(12-2))/12
a = 1800/12
a = 300/2
a = 150°
a is the single interior angle of the regular dodecagon.
b = 360-150-90
b = 360-240
b = 120°
(6√(3))² = 2c²-2c²cos120
108 = 3c²
c² = 36
c = 6 units.
c is the side length of the regular dodecagon, the square and the regular hexagon.
d² = 2*6²
d² = 72
d = √(72)
d = 6√(2) units.
d is the diagonal of the square.
e = ½(d)
e = ½(6√(2))
e = 3√(2) units.
f² = (3√(2))²+6²-2*3√(2)*6cos(105)
f = 8.1961524227 units.
g = 2(90-½(150))
g = 2(90-75)
g = 2*15
g = 30°
h = 90-30
h = 60°
tan60 = j/8.1961524227
j = 14.1961524227 units.
Therefore;
Painted area (area purple triangle) is;
½(fj)
= ½(8.1961524227*14.1961524227)
= 58.1769145362 square units.
= 58.18 square units to 2 decimal places.
