Mathematics Question and Solution
Let the radius of the two equal inscribed semi circles be 1 unit each.
2² = a²+1²
a² = 3
a = √(3) units.
b = 1+1+a
b = (2+√(3)) units.
b is the height of the ascribed right-angled triangle.
It implies, observing similar right-angled triangle side length ratios.
c - 1
(2+√(3)) - √(3)
Cross Multiply.
√(3)c = 2+√(3)
c = (2+√(3))/√(3)
c = ⅓(2√(3)+3)
c = 2.1547005384 units.
c is the base of the ascribed right-angled triangle.
Therefore, shaded area (blue area) fraction is;
(Area triangle with height and base ⅓(2√(3)+3) units and (2+√(3)) units respectively - Area circle with radius 1 unit) ÷ Area triangle with height and base ⅓(2√(3)+3) units and (2+√(3)) units respectively
= (½*⅓(2√(3)+3)*(2+√(3))-π(1)²)÷(½*⅓(2√(3)+3)*(2+√(3)))
= (⅙(4√(3)+6+6+3√(3))-π)÷(⅙(4√(3)+6+6+3√(3))
= (⅙(7√(3)+12)-π)÷(⅙(7√(3)+12))
= ⅙(7√(3)+12-6π)÷⅙(7√(3)+12)
= (7√(3)+12-6π)÷(7√(3)+12)
= 5.2747997314÷24.124355653
= 0.2186503883
