Mathematics Question and Solution
Let the radius of the ascribed circle be 1 unit.
Area ascribed circle is;
πr²
= π(1)²
= π square units.
Calculating area inscribed red circle.
Let it's radius be r.
sin30 = a/1
a = ½ units.
a = 0.5 units.
It implies;
(1-r)² = r²+(r+0.5)²
1-2r+r² = r²+r²+r+0.25
r²+3r-0.75 = 0
4r²+12r-3 = 0
Resolving the above quadratic equation via completing the square approach to get r, radius of the inscribed red circle.
r²+3r = ¾
(r+(3/2))² = ¾+(3/2)²
(r+(3/2))² = 3
r = -(3/2)±√(3)
It implies;
r = ½(2√(3)-3)
r = 0.2320508076 units.
Area inscribed red circle is;
πr²
= π(½(2√(3)-3))²
= ¼(12-12√(3)+9)π
= ¼(21-12√(3))π square units.
Therefore;
Red Inscribed Circle ÷ Ascribed Square is;
¼(21-12√(3))π÷π
= ¼(21-12√(3))
= 0.0538475773
