Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the side of the regular pentagon be 1 unit.
It implies;
Radius large circle, R is;
= 0.580673 unit.
Radius small circle, r is;
= 0.168821 unit.
(a) Radius Small Circle ÷ Radius Large Circle to 3 decimal places is;
0.168821 ÷ 0.580673
= 0.29073333873
≈ 0.291
Calculating pentagon area.
½(5)*(1/(2tan(180/5)))
= 1.72047740059 square units.
Shaded Area is;
Area triangle with height 1 unit and sin108 units - Area circle with radius 0.168821 unit - Area sector with radius 1.661347 units angle 58.28252558808° + Area triangle with height 1.661347 units and base 1.661347sin58.28252558808 units.
= ½*sin108 - π(0.168821)² - (58.28252558808*π*1.661347²÷360) + (½*1.661347²*sin58.28252558808)
= 0.47552825815 - 0.0895370558 - 1.40380387885 + 1.17392952768
= 0.15611685118 square units.
(b) Area Regular Pentagon ÷ Area Shaded to 1 decimal place is;
1.72047740059 ÷ 0.15611685118
= 11.02044646421
≈ 11
