/*! 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 32 Find the area of a regular dodec... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 32

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

Short Answer

Expert verified
The area of the regular dodecagon whose vertices are on the unit circle is approximately \(2.9963\) square units.

Step by step solution

01

Divide the dodecagon into 12 congruent triangles.

We will divide the dodecagon into 12 congruent triangles, each with a central angle of 30掳 (360掳/12=30掳).
02

Calculate the area of one triangle.

Using trigonometry, we can find the area of one equilateral triangle as follows: \[ A_{triangle} = \frac{1}{2} \cdot base \cdot height \] The base is equal to the length of each side of the triangle, which is equal to 2 times the length of the radius of the unit circle times the sine of half the central angle (15掳). The height is equal to the radius of the unit circle times the cosine of half the central angle (15掳). So the formula for the area of the triangle will be: \[ A_{triangle} = \frac{1}{2} \cdot (2 \cdot 1 \cdot \sin{15掳}) \cdot (1 \cdot \cos{15掳}) \]
03

Calculate the area of the dodecagon.

Now that we know the area of one triangle, we can find the total area of the dodecagon by multiplying the area of one triangle by 12: \[ A_{dodecagon} = 12 \cdot A_{triangle} = 12 \cdot \frac{1}{2} \cdot (2 \cdot 1 \cdot \sin{15掳}) \cdot (1 \cdot \cos{15掳}) \] Simplifying, we get: \[ A_{dodecagon} = 12 \cdot \sin{15掳} \cdot \cos{15掳} \]
04

Solve for the area of the dodecagon.

Now, use a calculator to find the numeric values of the trigonometric functions and solve for the area of the dodecagon: \[ A_{dodecagon} = 12 \cdot (0.2588) \cdot (0.9659) \approx 2.9963 \] The area of the regular dodecagon whose vertices are on the unit circle is approximately 2.9963 square units.

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Most popular questions from this chapter

The next two exercises emphasize that \(\sin (x-y)\) does not equal \(\sin x-\sin y .\) For \(x=79^{\circ}\) and \(y=33^{\circ}\), evaluate each of the following: (a) \(\sin (x-y)\) (b) \(\sin x-\sin y\)

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