Mathematics Question and Solution
Let the radius of the bigger green quarter circle be 1 unit.
Let the radius of the smaller green quarter circle be a.
Let the radius of the circle be b.
It implies;
b² = 1²+a²-2acos45
b² = 1+a²-√(2)a
b = √(1+a²-√(2)a) units.
c² = 2-2cos45
c = √(2-√(2)) units.
d = ½(1-a) units.
e = d+a
e = ½(1-a)+a
e = ½(1-a+2a)
e = ½(1+a) units.
Calculating a.
c² = e²+d²
√(2-√(2))² = (½(1+a))²+(½(1-a))²
2-√(2) = ¼(1+2a+a²)+¼(1-2a+a²)
2-√(2) = ¼(2+2a²)
4-2√(2) = 1+a²
a² = 3-2√(2)
a = √(3-2√(2)) units.
a = 0.4142135624 units.
a is radius of the smaller green quarter circle.
Therefore;
b = √(1+a²-√(2)a)
b = √(1+0.4142135624²-√(2)*0.4142135624)
b = 0.7653668647 units.
b is the radius of the circle.
It implies, area half circle is;
0.5πb²
= 0.5π(0.7653668647)²
= 0.9201511845 square units.
Area green (the two quarter circles area) is;
¼(1)²π+¼(0.4142135624)²π
= 0.7853981634+0.1347530211
= 0.9201511845 square units.
It implies;
Area green (the two quarter circles area) = half the circle's area
Proved!
