Mathematics Question and Solution
Calculating shaded area considering semi circle PQ.
Diameter of semi circle PQ will be 10cm, therefore radius is 5cm.
Shaded area will be;
Area semi circle PQ of radius 5cm and angle 180° - 2(area sector of radius 5cm and 60° - area equilateral triangle of equal side lengths 5cm each and angle 60° each angle) - 2(area sector of radius 5cm and 60° - area equilateral triangle of equal side lengths 5cm each and angle 60° each angle).
=(5²π)/2 - (2(5²π*60)/360 - (5²sin60)/2) - (2(5²π*60)/360 - (5²sin60)/2)
=25π/2 - (2(25π)/6 - (25√(3))/4) - (2(25π)/6 - (25√(3))/4)
=25π/2 - 25π/3 + 25√(3)/4 - 25π/3 + 25√(3)/4
=25π/2 - 50π/3 + 25√(3)/2
=(75π - 100π + 75√(3))/6
=(75√(3) - 25π)/6
=25(3√(3) - π)/6 cm²
= 8.561 cm²
Therefore shaded area is;
8.561 cm².
A smart approach is;
Shaded area will be;
2(area quarter circle of radius 5cm - 2(area sector of radius 5cm and angle 60°) + area equilateral triangle of lengths 5cm each side and angle 60°).
=2(25π/4 -2(25π)/6 + (25sin60)/2)
=2(25π/4 - 25π/3 + 25√(3)/4)
=25π/2 - 50π/3 + 25√(3)/2
=(75π - 100π + 75√(3))/6
=(75√(3) - 25π)/6
=25(3√(3) - π)/6 cm²
= 8.561 cm²
Therefore shaded area is;
8.561 cm².
