How to Find the Area of a Circle? Inside 4 Circles

Geometry math problem

Figure shows 4 circles with a radius of 10 cm. The blue circle is tangent to all four circles then find the area of small circle

Solution: Area of the circle

We can connect the center and form a triangle as shown in the figure, where circles are tangential so the angle BAC is tangential (∠BAC = 90°)

From the figure

AB = 20 cm, AC = 20 cm, BQ = 10 cm and PC = 10 cm

Let PQ = 2r = Diameter of the blue circle

To find the area of the circle we need to find the radius of the circle

Apply Pythagoras theorem in triangle ABC

BD² = AB² + AD²

BD² = 20² + 20² = 2×20²

BD = 20 √2

From the figure, we can also get

PQ = BC – 20

PQ = 20√2 – 20 = 20(√2 – 1) cm

So the radius of small circle = 10√2 – 10 cm

Area of the circle = π(10√2 – 10)² = π(200 – 200√2 + 100) = 300π – 200π√2 cm²