four circles of radious rCM are place such away that their centres from a square

 four circles of radious rCM are place such away that their centres from a square. find the ratio of the area of the square (formed by the centres of the circles) to the area of the rhombus formed by the touching points of the circle

Let's break down the problem step by step:

**Given:**  

- Four circles of radius **r** are placed such that their centres form the vertices of a square.

- These circles touch each other.

**You are to find:**  

The ratio of the area of the square (formed by the centres) to the area of the rhombus (formed by the points where the circles touch).

***

## 1. **Area of the Square Formed by Centres**

Let the side of the square be **a**.

Since the circles of radius **r** just touch each other at the sides, the centres of circles that are adjacent are **2r** apart.

So,  

**Side of the square, $$ a = 2r $$**

- **Area of the square = $$ a^2 = (2r)^2 = 4r^2 $$**

***

## 2. **Area of Rhombus Formed by Touching Points**

**Where are the touching points?**

- The touching points on sides of the square are the midpoints, since distance from a centre to the touching point along the side = r.

- So, the four touching points are:

    - Midpoint of left side,

    - Midpoint of right side,

    - Midpoint of top side,

    - Midpoint of bottom side.

If the square's centres are at  

$$ (0, 0), (2r, 0), (2r, 2r), (0, 2r) $$, then the touching points are at:  

- $$ (r, 0) $$  

- $$ (2r, r) $$  

- $$ (r, 2r) $$  

- $$ (0, r) $$

These touching points form a **rhombus** inside the square.

**Diagonals of the rhombus:**

- One diagonal: Between $$ (r, 0) $$ and $$ (r, 2r) $$, length = $$ 2r $$

- Other diagonal: Between $$ (0, r) $$ and $$ (2r, r) $$, length = $$ 2r $$

So both diagonals = $$ 2r $$

- **Area of a rhombus = $$ \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 2r \times 2r = 2r^2 $$**

***

## 3. **Ratio**

$$

\text{Ratio} = \frac{\text{Area of Square}}{\text{Area of Rhombus}} = \frac{4r^2}{2r^2} = 2:1

$$

***

### **Final Answer**

$$

\boxed{2:1}

$$

The ratio of the area of the square (formed by the centres) to the area of the rhombus (formed by the touching points) is **2:1**.