Question 4: Measure of Each Side of the Hexagon
To find the side length of a regular hexagon given the area and the apothem, we can use the formula for the area of a regular polygon:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]
For a regular hexagon, the perimeter (P) can be expressed as:
\[ P = 6 \times s \quad \text{(where s is the side length)} \]
Substituting this into the area formula gives:
\[ \text{Area} = \frac{1}{2} \times (6s) \times a \]
Where \(a\) is the apothem. Given that the area is 516 square meters and the apothem is 16 meters, we can set up the equation:
\[ 516 = \frac{1}{2} \times (6s) \times 16 \]
Simplifying this equation:
\[ 516 = 48s \]
Now, solving for \(s\):
\[ s = \frac{516}{48} \approx 10.75 \]
Thus, the side length of the hexagon is approximately:
\[ \text{The hexagon has side lengths of} \ 10.75 \ \text{meters.} \]
Question 5: Area of the Regular 15-Sided Polygon
To find the area of a regular polygon, we can use the same area formula:
\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \]
For a 15-sided polygon, the perimeter (P) is:
\[ P = 15 \times s \quad \text{(where s is the side length)} \]
Given that the side length \(s = 9\) mm and apothem \(a = 12\) mm, we can calculate the perimeter:
\[ P = 15 \times 9 = 135 \text{ mm} \]
Now we can find the area:
\[ \text{Area} = \frac{1}{2} \times 135 \times 12 \]
Calculating the area:
\[ \text{Area} = \frac{1}{2} \times 135 \times 12 = \frac{1620}{2} = 810 \text{ mm}^2 \]
Thus, the area of the regular 15-sided polygon is:
\[ \text{The area of the regular 15-sided shape is} \ 810 \ \text{mm}^2. \]
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