# 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