To solve this problem, we need to calculate the area of the rectangle containing the three circles. Let's break down the solution step by step:

Step 1: Understanding the arrangement

• The three circles are identical, each with a radius of 2 mm.
• Each circle touches the other two circles and the sides of the rectangle.
• This suggests that two circles are positioned side by side horizontally, while the third circle is directly beneath or above these two.

Step 2: Dimensions of the rectangle

1. Horizontal Length of the Rectangle:

   • The horizontal arrangement has two circles touching side by side.
   • Since each circle has a radius of 2 mm, their diameter is $2 \times 2 = 4$ mm.
   • Placing two circles side by side, their total length becomes $4 + 4 = 8$ mm.

2. Vertical Height of the Rectangle:

   • The third circle is positioned below or above the pair of circles. The distance between the centers of the circles that are diagonally arranged forms an equilateral triangle.
   • The height of an equilateral triangle with side length $4$ mm (the distance between the centers of the circles) is $\frac{\sqrt{3}}{2} \times 4 = 2\sqrt{3}$ mm.
   • Adding the radius of the circle (2 mm) at the top and bottom, the total vertical height of the rectangle becomes: $2 + 2\sqrt{3} + 2 = 4 + 2\sqrt{3} \text{ mm}$

Step 3: Area of the rectangle

• The length of the rectangle is $8$ mm.
• The height of the rectangle is $4 + 2\sqrt{3}$ mm.
• The area of the rectangle is calculated as: $\text{Area} = 8 \times (4 + 2\sqrt{3}) = 32 + 16\sqrt{3} \text{ mm}^2$

Final Answer

The exact area of the rectangle is $16\sqrt{3} + 32$ mm$^2$.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How is the height of an equilateral triangle derived?
2. Why do we multiply the height by $2\sqrt{3}$ instead of $4\sqrt{3}$?
3. What if the radius of each circle was different? How would that change the calculations?
4. How would you calculate the perimeter of this rectangle?
5. Can you find the area of the circles and the area that remains unoccupied inside the rectangle?

Tip: For problems involving touching circles, it’s often helpful to visualize the arrangement as a geometric shape like triangles to determine dimensions.