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
![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](https://aplusreach.com/wp-content/uploads/2021/12/How-to-Find-the-Area-of-a-Circle-Tangential-to-4-Circle-1-1024x576.jpg)
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°)
![](https://aplusreach.com/wp-content/uploads/2021/12/How-to-Find-the-Area-of-a-Circle-Tangential-to-4-Circle-2-1024x576.jpg)
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](https://aplusreach.com/wp-content/uploads/2021/12/How-to-Find-the-Area-of-a-Circle-Tangential-to-4-Circle-3-1024x576.jpg)
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