/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Find the perimeter of a regular ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 34

Find the perimeter of a regular dodecagon whose vertices are on the unit circle.

Short Answer

Expert verified
The perimeter of the regular dodecagon with vertices on the unit circle is 12\(\sqrt{\frac{8+8 \sqrt{6} + 2\sqrt{2}}{4}}\).

Step by step solution

01

Visualize the dodecagon on the unit circle

To begin, imagine the unit circle, with a regular dodecagon inscribed within it (with vertices on the unit circle). A regular dodecagon has 12 equal sides and an interior angle of 150 degrees.
02

Get half of the angle

Since we will be using trigonometry to find the side length, we can use the half of the interior angle formed by a vertex of the dodecagon. The interior angle of a regular dodecagon is \(150^\circ\), so half the interior angle is \(75^\circ\).
03

Use the law of cosines to find the side length

Since the vertices of the dodecagon lie on the unit circle, the distance from the center of the circle to a vertex is 1 unit. By forming an isosceles triangle with two radii and a side of the regular dodecagon, we can use the law of cosines: Let s be the side length of the dodecagon: \(s^2 = 1^2 + 1^2 - 2(1)(1)(\cos{150^\circ})\) Together with the half angle, we substitute the angle and get: \(s^2 = 1^2 + 1^2 - 2(1)(1)(\cos{(2)(75^\circ)})\)
04

Evaluate the expression

Evaluate the expression from Step 3 to solve for the side length, s: \(s^2 = 1^2 + 1^2 - 2(1)(1)(\cos{150^\circ})\) \(s^2 = 1 +1- 2(\cos{150^\circ})\) \(s^2 = 2 - 2(-\frac{\sqrt{6}+\sqrt{2}}{4})\) \(s^2 = 2+2(\frac{\sqrt{6}+\sqrt{2}}{4})\) Put everything under the same denominator: \(s^2 = \frac{8+8 \sqrt{6} + 2\sqrt{2}}{4}\) Taking the square root of both sides, we find the side length of the dodecagon: \(s = \sqrt{\frac{8+8 \sqrt{6} + 2\sqrt{2}}{4}}\)
05

Calculate the perimeter

Now that we have the side length, s, we can find the perimeter of the dodecagon by multiplying the side length by the number of sides, which is 12: Perimeter = 12s Perimeter = 12\(\sqrt{\frac{8+8 \sqrt{6} + 2\sqrt{2}}{4}}\) Hence, the perimeter of the regular dodecagon with vertices on the unit circle is 12\(\sqrt{\frac{8+8 \sqrt{6} + 2\sqrt{2}}{4}}\).

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