Mathematics Question and Solution
Sir Mike Ambrose is the author of the question.
Let the square side be 2 units.
Area Square is;
2²
= 4 square units.
Calculating r, radius of the inscribed circle.
√(2)r+½√(2)r+½√(10)r = ½√(2)*√(2)
r = ¼(3√(2)-√(10)) units.
Area Shaded is;
Area triangle with height √(2) units and base ½√(2) unit - Area circle with radius ¼(3√(2)-√(10)) unit.
= ½*√(2)*½√(2) - π(¼(3√(2)-√(10)))²
= ½ - ¼*π(7-3√(5))
= ¼(2-7π+3√(5)π) square units.
It implies;
Area Shaded ÷ Area Square exactly in terms of π as a single fraction in its simplest form is;
¼(2-7π+3√(5)π) ÷ 4
= (2-7π+3√(5)π)/16
