How do you find the shaded area of three tangent circles in a rectangle?

Expert intro: This is a classic packing problem. The clean move is to turn the circle arrangement into rectangle dimensions first, then subtract the total circle area from the rectangle area.

Detailed walkthrough

Step 1: Read the circle arrangement

There are 3 identical circles, each with radius $1$ m. Because each circle touches the other two, their centers form an equilateral triangle with side length $2$.

Step 2: Find the rectangle dimensions

Since the circles also touch the sides of the rectangle, the width is the diameter of one circle times 3:

$\text{width}=3\times 2=6$

For the height, the three circle centers stack in a way that gives one radius above and one radius below the center line, plus the vertical spacing from the equilateral triangle. The height is:

$\text{height}=2+2\cdot \frac{\sqrt{3}}{2}=2+\sqrt{3}$

So the rectangle area is:

$A_{\text{rect}}=6(2+\sqrt{3})=12+6\sqrt{3}$

Step 3: Find the total area of the 3 circles

Each circle has area:

$\pi r^2=\pi(1)^2=\pi$

So all three circles have area:

$A_{\text{circles}}=3\pi$

Step 4: Subtract to get shaded area

If the shaded region is the part of the rectangle outside the circles, then:

$A_{\text{shaded}}=(12+6\sqrt{3})-3\pi$

Final answer

$\boxed{12+6\sqrt{3}-3\pi}$

💡 Pitfall guide

A common error is adding the circle areas to the rectangle area instead of subtracting them. Another one is using diameter $=1$ when the radius is $1$ m; the diameter is $2$ m. In exact-form problems, keep $\pi$ and $\sqrt{3}$ in the answer rather than turning them into decimals too early.

🔄 Real-world variant

If the shaded region were only the spaces between the circles, you would not subtract the entire rectangle area. You would instead identify the curvy gaps and use sector areas or symmetry. Changing the circle radius would scale both the rectangle dimensions and the circle areas, so the same method still works with a different number plugged in.

🔍 Related terms

tangent circles, shaded area, exact form