Given three equal circles that are placed inside an equilateral triangle such that every circle is tangential to the sides of the equilateral triangle and to other circles. The task is to find the ratio of the area of one circle to the area of the equilateral triangle.

**Solution:**

Below is the image of how circles are inscribed in a triangle:

Now since AB and BC are tangents to the circle with center P, PQ will be perpendicular to BC and PB will bisect angle ABC. Hence angle PBQ=30° since ABC is an equilateral triangle and angle ABC=60°.

Consider the triangle PBQ, tan30°= PQ/BQ = 1/√3

BQ = PQ*√3 = R*√3 (R is radius one circle). Similarly RC = R*√3

Now BC = BQ+QR+CR = R√3 + 2R + R√3 = 2R(√3 +1)

Therefore, the ratio of the area of the circle to the area of the triangle is given by:

Since,

[Tex]area(triangle) = \frac{\sqrt{3}(2r(\sqrt{3}+1))^{2}}{4} [/Tex]

Therefore, the ratio is given by:

[Tex]Ratio = \frac{\pi}{\sqrt{3}(\sqrt{3}+1)^{2}} [/Tex]

## Recommended Posts:

- Puzzle | Form Three Equilateral Triangles
- Puzzle | Connect 9 circles each arranged at center of a Matrix using 3 straight lines
- Equilateral Triangles using Matchsticks
- Count of Equilateral Triangles of unit length possible from a given Hexagon
- Puzzle 17 | (Ratio of Boys and Girls in a Country where people want only boys)
- Puzzle 63 | Paper ball and three friends
- Puzzle | Three Aliens Riddle
- Puzzle | Three Thief crossing river
- Puzzle | The Three Pile Trick
- Puzzle | Three Brothers Travelled by Road
- Puzzle | Three Squares
- Puzzle 81 | 100 people in a circle with gun puzzle
- Puzzle | The Circle of Lights
- Puzzle | Place numbers 1 to 9 in a Circle such that sum of every triplet in straight line is 15
- Puzzle | Maximum pieces that can be cut from a Circle using 6 straight lines
- Area of squares formed by joining mid points repeatedly
- Puzzle 21 | (3 Ants and Triangle)
- Puzzle 67 | Fit Triangle
- Puzzle | One Mile on the Globe
- Puzzle | 3 cuts to cut round cake into 8 equal pieces

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.