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