Mathematics Question and Solution
Let the side length of the regular hexagon be 1 unit.
a = ½(180-108)
108° = single interior angle of the regular pentagon.
180° = sum of angles on a straight line.
a = ½(72)
a = 36°
tan36 = b/0.5
b = 0.363271264 units.
Therefore, area blue is;
½(0.5a*b)
= ½(0.5*0.363271264)
= 0.090817816 square units.
Calculating Lilac Area.
Let the side length of the regular pentagon be c.
d = ½(c) units.
e² = 2-2cos120
e = √(3) units.
f² = 2c²-2c²cos108
f = 1.6180339887c units.
Calculating c.
f² = e²+d²
(1.6180339887c)² = √(3)²+(0.5c)²
2.6180339887c²-0.25c² = 3
2.3680339887c² = 3
c = 1.1255548445 units.
Again, c is the side length of the regular.
sin54 = g/h
g is the radius of the inscribed lilac circle.
h = g/(sin54)
h = 1.2360679775g units.
It implies, calculating g.
h+g+1 = √(3)
1.2360679775g+g = √(3)-1
2.2360679775g = (√(3)-1)
g = 0.3273830737 units.
Again, g is the radius of the inscribed lilac circle.
Area Lilac Circle is;
π(0.3273830737)²
= 0.3367148858 square units.
Therefore;
Area Blue ÷ Area Lilac to 2 decimal places is;
0.090817816÷0.3367148858
= 0.2697172589
≈ 0.27 to 2 decimal places.
