Question

A regular hexagon is below. Solve for the area of the hexagon.
The area of a regular hexagon with side length of 14.

A watch has the SAME hexagonal face as the picture to the left. If the radius of the circle is 4, then what is the area between the hexagon and circle?

Answer 1

To find the area of a regular hexagon with a side length of 14, we can use the formula for the area of a regular hexagon:

\[
\text{Area} = \frac{3\sqrt{3}}{2} s^2
\]

where \( s \) is the length of a side.

Plugging in \( s = 14 \):

\[
\text{Area} = \frac{3\sqrt{3}}{2} \times 14^2
\]
\[
= \frac{3\sqrt{3}}{2} \times 196
\]
\[
= 294\sqrt{3}
\]

Now we will calculate the area of the circle with a radius of 4. The formula for the area of a circle is:

\[
\text{Area} = \pi r^2
\]

where \( r \) is the radius. Now plug in \( r = 4 \):

\[
\text{Area} = \pi \times 4^2
\]
\[
= \pi \times 16
\]
\[
= 16\pi
\]

Finally, we need to find the area between the hexagon and the circle. This is simply the area of the circle minus the area of the hexagon:

\[
\text{Area between} = \text{Area of Circle} - \text{Area of Hexagon}
\]
\[
= 16\pi - 294\sqrt{3}
\]

Thus, the area between the hexagon and the circle is:

\[
16\pi - 294\sqrt{3}
\]