How to find area covered by a car windshield wiper when it swaps a certain angle?

enter image description here

The area we want to find out is the one surrounded by red in the figure above. This is the worst windshield wiper ever.

To verify this, we know that since $\overline{AE}\perp\overline{MN}$, the inner circle — with center $M$ and radius $\overline{MN}$ — will never be reached. Same with the two blue areas at the left and right bottom corner.

I assume that it was the inaccurate figure you have that misled you. Anyway, here's my solution. The length and area unit below are $m$ and $m^2$, respectively.

Let's find out the length of $\overline{MN}$ first by noting that $\triangle ANM$ is a right triangle with $\angle AMN=30^{\circ}$. Therefore $$\overline{AM}=0.5\Longrightarrow \overline{MN}=\frac{\sqrt3}4$$

Now let's compute the red area. Let it be $S$.

$$\begin{align}S&=\frac{1}2\cdot (0.5)^2\pi-\frac{120}{360}\cdot \left(\frac{\sqrt3}4\right)^2\pi-2\triangle ANM\\ &=\frac{\pi}8-\frac{\pi}{16}-2\cdot \frac{1}2\cdot \frac{1}4\cdot\frac{\sqrt3}4\\&=\color{red}{\frac{\pi}{16}-\frac{\sqrt3}{16}} \end{align}$$

which is the final answer.

Tags:

Euclidean Geometry

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