Question

o is the center of the regular dodecagon below. find its perimeter. round to the nearest tenth if necessary.

Explanation:

Step1: Find central angle

A regular dodecagon has 12 sides. The central angle for each side is $\frac{360^\circ}{12} = 30^\circ$.

Step2: Split triangle, find half-side

The radius (distance from center to a vertex) is 4. If we split the isosceles triangle (formed by two radii and a side) into two right triangles, the angle at the center for each right triangle is $\frac{30^\circ}{2}=15^\circ$. Let the length of half the side be $s/2$. Using sine: $\sin(15^\circ)=\frac{s/2}{4}$, so $s/2 = 4\sin(15^\circ)$.

Step3: Calculate side length

$\sin(15^\circ)\approx0.2588$, so $s/2 = 4\times0.2588 = 1.0352$, then side length $s = 2\times1.0352 = 2.0704$.

Step4: Calculate perimeter

Perimeter $P = 12\times s = 12\times2.0704 = 24.8448\approx24.8$ (rounded to nearest tenth) or more accurately, using exact trigonometric values: $\sin(15^\circ)=\frac{\sqrt{6}-\sqrt{2}}{4}$, so $s = 2\times4\times\frac{\sqrt{6}-\sqrt{2}}{4}=2(\sqrt{6}-\sqrt{2})\approx2(2.449 - 1.414)=2(1.035)=2.07$, then perimeter $12\times2.07 = 24.84\approx24.8$ (or more precisely, using the law of cosines on the isosceles triangle: $s^2 = 4^2 + 4^2 - 2\times4\times4\times\cos(30^\circ)=32 - 32\times\frac{\sqrt{3}}{2}=32 - 16\sqrt{3}\approx32 - 27.7128 = 4.2872$, so $s\approx2.0705$, perimeter $12\times2.0705\approx24.8$).

Answer:

The perimeter of the regular dodecagon is approximately $\boxed{24.8}$ (or more accurately, if we calculate with more precision, using $s = 2\times4\times\sin(15^\circ)$, perimeter $12\times8\times\sin(15^\circ)=96\times\sin(15^\circ)\approx96\times0.2588 = 24.8448\approx24.8$). Wait, actually, a better way: the formula for the side length of a regular polygon with radius $r$ (distance from center to vertex) is $s = 2r\sin(\frac{\pi}{n})$, where $n = 12$. So $s = 2\times4\times\sin(\frac{\pi}{12})$, $\sin(\frac{\pi}{12})=\sin(15^\circ)\approx0.2588$, so $s\approx2\times4\times0.2588 = 2.0704$, perimeter $12\times2.0704 = 24.8448\approx24.8$ (to nearest tenth) or $24.8$ (if we consider the exact calculation, but sometimes using the law of cosines: $s^2 = r^2 + r^2 - 2r^2\cos(\frac{2\pi}{n}) = 2r^2(1 - \cos(\frac{2\pi}{12})) = 2r^2(1 - \cos(\frac{\pi}{6}))$. $r = 4$, so $s^2 = 32(1 - \frac{\sqrt{3}}{2}) = 32 - 16\sqrt{3}\approx32 - 27.7128 = 4.2872$, $s\approx2.0705$, perimeter $12\times2.0705\approx24.84\approx24.8$). So the final answer is $\boxed{24.8}$.