Geometry Math Problem: Find the Relation between Area of Circle and Semicircle Inside the Square
Find the Relation between Area of Circle and Semicircle Inside a Square
From the figure, ABCD is a square and CT is the tangent of the semicircle and circle. Then find the Relation between the area of the semicircle and the area of the circle inside the square

Solution
Connect C and the centre of the semicircle

Let’s assume the radius of the semicircle is R and the radius of the circle is r, then
Area of red semicircle = πR²/2
Area of blue circle = πr²
Now, the area of blue circle / area of red semicircle = (πr²)/(πR²/2)
⇒ Area of blue circle / area of red semicircle = 2r²/R²
Apply Pythagoras theorem in the triangle OBC
OC² = OB² + BC²
⇒ OC² = R² + (2R)²
⇒ OC² = R² + 4R² = 5R²
Thus, OC = R√5
From the figure triangle OBC and triangle PQC are similar, so
OB/PR = OC/PC
⇒ R/r = (R√5)/(R√5 – (R + r))
⇒ R(R√5 – (R + r)) = (R√5)r
so, R√5 – (R + r) = r√5
⇒ R(√5 – 1) = r(√5 +1)
⇒ r/R = (3 – √5)/2
Square both sides, then
r²/R² = (3 – √5)² / 4 = (14 – 6√5) / 4
⇒ r²/R² = (7 – 3√5) / 2
⇒ 2r²/R² = 7 – 3√5
Area of blue circle / area of red semicircle = 7 – 3√5