Chapter 29: Chapter 29

Problem 555

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If the length of a side of a regular polygon of 12 sides is represented by s, and the length of its apothem is represented by a , find the area of the polygon, \(\mathrm{A}\), in terms of a and \(\mathrm{s}\).

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The area of the regular 12-sided polygon in terms of s and a is: \[\mathrm{A} = 6 s a\]

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Step 1: Find the Central Angle

To determine the area of the polygon, we must first find the central angle that encloses each of the 12 individual triangles that make up the polygon. Since the polygon is regular, all 12 central angles are equal. To find one of these angles, we can divide the total angle measurement of the polygon (360 degrees) by the number of sides (12): \[ \text{Central angle } (\theta) = \frac{360^\circ}{12} = 30^\circ \]

Step 2: Find the Base and Height of Each Triangle

Now that we know the central angle, we can find the base and height of each of the 12 individual triangles to calculate their areas. The base of each triangle is simply the side length of the polygon, represented by s. The height of each triangle is the apothem of the polygon, represented by a. Each triangle formed is an isosceles triangle, with the apothem serving as the height and the central angle bisected, creating two 30-60-90 triangles.

Step 3: Calculate Half of the Central Angle

To work with the 30-60-90 triangles, we need to find half of the central angle, which is: \[\frac{\theta}{2} = \frac{30^\circ}{2} = 15^\circ\]

Step 4: Find the Area of One Triangle

Now that we have the base and height of one triangle and the half of the central angle, we can use the formula for the area of a triangle: \[ \text{Area of one triangle} = \frac{1}{2}\text{ base } \times \text{ height} = \frac{1}{2} s \times a \]

Step 5: Calculate the Area of the Entire Polygon

Finally, we can find the area of the entire polygon by multiplying the area of one triangle by the number of sides in the polygon (12): \[\mathrm{A} = 12 \times \frac{1}{2} s \times a = 6 s a\] Thus, the area of the regular 12-sided polygon in terms of s and a is: \[\mathrm{A} = 6 s a\]

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Chapter 29: Chapter 29

If the length of a side of a regular polygon of 12 sides is represented by s, and the length of its apothem is represented by a , find the area of the polygon, \(\mathrm{A}\), in terms of a and \(\mathrm{s}\).

Short Answer

Step by step solution

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Step 1: Find the Central Angle

Step 2: Find the Base and Height of Each Triangle

Step 3: Calculate Half of the Central Angle

Step 4: Find the Area of One Triangle

Step 5: Calculate the Area of the Entire Polygon

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Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

Chapter 25

Chapter 26

Chapter 27

Chapter 28

Chapter 30

Chapter 31

Chapter 32

Chapter 33

Chapter 34

Chapter 35

Chapter 36

Chapter 37

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Chapter 39

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Chapter 41

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Chapter 43

Chapter 44

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Chapter 51

Chapter 52

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