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Problem 555

# 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 $$\mathrm{s}$$.

Expert verified
The area of the regular 12-sided polygon in terms of s and a is: $\mathrm{A} = 6 s a$
See the step by step solution

## 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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