Mathematics Question and Solution
Calculating Area Blue Plus Area Orange.
Radius of the congruent four semi circles is 1 unit each.
(a-1)²+1² = 2²
a²-2a+2 = 4
a²-2a-2 = 0
Resolving the above quadratic equation via completing the square approach to get a.
(a-1)² = 2+(-1)²
a-1 = ±√(3)
a = 1±√(3)
It implies;
a ≠ 1-√(3)
a = 1+√(3) units.
a = 2.7320508076 units.
a is the side length of the square.
Area Blue Plus Area Orange is;
Area square with side (1+√(3)) units - 4(area semi circle with radius 1 unit)
= (1+√(3))²-4(0.5π*1²)
= (4+2√(3)-2π) square units.
Calculating Orange Area.
tanb = √(3)/1
b = atan(√(3))
b = 60°
c = 90-b
c = 90-60
c = 30°
Area Orange is;
4(area triangle with height √(3) units and base 1 unit - Area quarter circle with radius 1 unit).
= 4((½*√(3)*1)-(¼*π*1²)
= 4(½*√(3)-¼*π)
= (2√(3)-π) square units.
Therefore, area Blue is;
Area Blue Plus Area Orange - Area Orange
= (4+2√(3)-2π)-(2√(3)-π)
= (4-π) square units.
It implies;
Blue Area - Orange Area is;
(4-π)-(2√(3)-π)
= (4-2√(3)) square units.
= 2(2-√(3)) square units.
= 0.5358983849 square units.
