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To determine which regular polygon with an apothem of 5 inches has a perimeter closest to the circumference of a circle with a diameter of 10.8 inches, we follow these steps:
1. Calculate the Circumference of the Circle:
- Given: Diameter = 10.8 inches
- The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = \pi \times \text{Diameter} \][/tex]
- Plugging in the diameter value:
[tex]\[ \text{Circumference} = \pi \times 10.8 \approx 33.93 \text{ inches} \][/tex]
2. Determine the Perimeter of Each Polygon:
- Given: Apothem = 5 inches
- The formula for the perimeter of a regular polygon is:
[tex]\[ \text{Perimeter} = 2 \times \text{number of sides} \times \text{apothem} \times \tan\left(\frac{\pi}{\text{number of sides}}\right) \][/tex]
- We calculate the perimeter for each of the following polygons: pentagon, square, octagon, and hexagon.
- Pentagon (5 sides):
[tex]\[ \text{Perimeter} = 2 \times 5 \times 5 \times \tan\left(\frac{\pi}{5}\right) \approx 36.33 \text{ inches} \][/tex]
- Square (4 sides):
[tex]\[ \text{Perimeter} = 2 \times 4 \times 5 \times \tan\left(\frac{\pi}{4}\right) = 40 \text{ inches} \][/tex]
- Octagon (8 sides):
[tex]\[ \text{Perimeter} = 2 \times 8 \times 5 \times \tan\left(\frac{\pi}{8}\right) \approx 33.14 \text{ inches} \][/tex]
- Hexagon (6 sides):
[tex]\[ \text{Perimeter} = 2 \times 6 \times 5 \times \tan\left(\frac{\pi}{6}\right) \approx 34.64 \text{ inches} \][/tex]
3. Compare the Perimeters to the Circumference:
- We need to find which polygon’s perimeter is closest to the circle's circumference of 33.93 inches.
- Pentagon: 36.33 inches
- Square: 40 inches
- Octagon: 33.14 inches
- Hexagon: 34.64 inches
4. Determine the Closest Match:
- Calculate the difference between each polygon's perimeter and the circle's circumference:
[tex]\[ \begin{align*} \text{Pentagon:} & \quad |36.33 - 33.93| = 2.4 \\ \text{Square:} & \quad |40 - 33.93| = 6.07 \\ \text{Octagon:} & \quad |33.14 - 33.93| = 0.79 \\ \text{Hexagon:} & \quad |34.64 - 33.93| = 0.71 \\ \end{align*} \][/tex]
- From these differences, we see that the hexagon’s perimeter (34.64 inches) is the closest to the circle’s circumference (33.93 inches).
Conclusion:
The regular polygon with an apothem of 5 inches whose perimeter is closest to the circumference of a circle with a diameter of 10.8 inches is the hexagon.
1. Calculate the Circumference of the Circle:
- Given: Diameter = 10.8 inches
- The formula for the circumference of a circle is:
[tex]\[ \text{Circumference} = \pi \times \text{Diameter} \][/tex]
- Plugging in the diameter value:
[tex]\[ \text{Circumference} = \pi \times 10.8 \approx 33.93 \text{ inches} \][/tex]
2. Determine the Perimeter of Each Polygon:
- Given: Apothem = 5 inches
- The formula for the perimeter of a regular polygon is:
[tex]\[ \text{Perimeter} = 2 \times \text{number of sides} \times \text{apothem} \times \tan\left(\frac{\pi}{\text{number of sides}}\right) \][/tex]
- We calculate the perimeter for each of the following polygons: pentagon, square, octagon, and hexagon.
- Pentagon (5 sides):
[tex]\[ \text{Perimeter} = 2 \times 5 \times 5 \times \tan\left(\frac{\pi}{5}\right) \approx 36.33 \text{ inches} \][/tex]
- Square (4 sides):
[tex]\[ \text{Perimeter} = 2 \times 4 \times 5 \times \tan\left(\frac{\pi}{4}\right) = 40 \text{ inches} \][/tex]
- Octagon (8 sides):
[tex]\[ \text{Perimeter} = 2 \times 8 \times 5 \times \tan\left(\frac{\pi}{8}\right) \approx 33.14 \text{ inches} \][/tex]
- Hexagon (6 sides):
[tex]\[ \text{Perimeter} = 2 \times 6 \times 5 \times \tan\left(\frac{\pi}{6}\right) \approx 34.64 \text{ inches} \][/tex]
3. Compare the Perimeters to the Circumference:
- We need to find which polygon’s perimeter is closest to the circle's circumference of 33.93 inches.
- Pentagon: 36.33 inches
- Square: 40 inches
- Octagon: 33.14 inches
- Hexagon: 34.64 inches
4. Determine the Closest Match:
- Calculate the difference between each polygon's perimeter and the circle's circumference:
[tex]\[ \begin{align*} \text{Pentagon:} & \quad |36.33 - 33.93| = 2.4 \\ \text{Square:} & \quad |40 - 33.93| = 6.07 \\ \text{Octagon:} & \quad |33.14 - 33.93| = 0.79 \\ \text{Hexagon:} & \quad |34.64 - 33.93| = 0.71 \\ \end{align*} \][/tex]
- From these differences, we see that the hexagon’s perimeter (34.64 inches) is the closest to the circle’s circumference (33.93 inches).
Conclusion:
The regular polygon with an apothem of 5 inches whose perimeter is closest to the circumference of a circle with a diameter of 10.8 inches is the hexagon.
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