Mathematics Question and Solution
Let the regular pentagon side be 1 unit.
sin54 = a/1
a = 0.80901699437 unit.
b = 2a
b = 1.61803398875 units.
Let the regular hexagon side be c.
Calculating c.
c+2(csin30) = b
c+2(0.5c) = 1.61803398875
c = 0.80901699437 unit.
Calculating Area Yellow.
It is;
Area regular pentagon with side 1 unit - Half Area regular hexagon with side 0.80901699437 unit - Area triangle with height 1 unit and base sin108 units.
= ½(5*1)*(1/(2tan(180/5))) - ½*½(0.80901699437²*6)*(1/(2tan(180/6))) - ½*1*sin108
= 1.72047740059 - 0.85023147833 - 0.47552825815
= 0.39471766411 square units.
Calculating Area Shaded.
sin36 = 0.5/d
d = 0.85065080835 unit.
e² = 0.85065080835²+0.80901699437²-2*0.80901699437*0.85065080835cos78
e = 1.04496382838 units.
Where e is the radius of the arc.
1.04496382838² = f²+0.85065080835²-2*f*0.85065080835co162
f²+1.61803398875f-0.36834260488 = 0
f = 0.202344 unit.
(0.202344/sing) = (1.04496382838/sin162)
g = 3.43046960466°
(0.80901699437/sinh) = (1.04496382838/sin78)
h = 49.22564879355°
i = g+h
i = 52.65611839821°
It implies;
Area Shaded is;
Area sector with radius 1.04496382838 units and angle 52.65611839821 - Area triangle with height 0.80901699437 unit and base 0.85065080835sin78 units - Area triangle with height 0.202344 unit and base 0.85065080835sin162
= 0.50176310994 - 0.3365761683 - 0.02659463404
= 0.1385923076 square units.
Therefore;
Area Shaded ÷ Area Yellow to 2 decimal places is;
0.1385923076 ÷ 0.39471766411
= 0.35111757137
= 0.35
