Mathematics Question and Solution
Let the side length of the regular pentagon be 1 unit.
a = ½(108)-25
a = 54-25
a = 29°
b = 72-29
b = 43°
(c/sin108) = (1/sin43)
c = 1.3945143742 units.
tan25 = d/1.3945143742
d = 0.6502727321 units.
Area Yellow is;
0.5*1*0.6502727321sin25
= 0.0687042829 square units.
Calculating Shaded Area.
e = 54+25
e = 79°
f² = 1²+1.3945143742²-2*1.3945143742cos79
f = 1.5532219952 units.
(1.5532219952/sin79) = (1/sing)
g = 39.1973995412°
h = 39.1973995412+25
h = 64.1973995412°
j = 180-64.1973995412
j = 115.8026004588°
(k/sin25) = (1.3945143742/sin115.8026004588)
k = 0.6546127784 units.
l = 1.5532219952-0.6546127784
l = 0.8986092168 units.
l is the radius of the shaded arc.
m = 180-2*64.1973995412
m = 51.6052009176°
Area Shaded is;
Area sector with angle 51.6052009176°and radius 0.8986092168 units - Area triangle with height 0.8986092168 units and base 0.8986092168sin51.6052009176 units.
(51.6052009176π*0.8986092168²/360)-0.5*0.8986092168²*sin51.6052009176
= 0.3636491549-0.3164384187
= 0.0472107362 square units.
It implies;
Area Shaded ÷ Area Yellow to 2 decimal places is;
0.0472107362÷0.0687042829
= 0.6871585608
≈ 0.69
