Which Of The Following Measures Of Central Angles Yield A Regular Polygon?A. 6° B. 14° C. 80° D. 40° 15. Find The Number Of Sides Of A Regular Polygon Whose Measure Of The Central Angle Is 12°.16. Find The Perimeter, Area, And Apothem Of A
Understanding Central Angles in Polygons
In geometry, a central angle is an angle whose vertex is at the center of a circle, and its sides pass through the endpoints of an arc. When it comes to regular polygons, the measure of central angles plays a crucial role in determining the number of sides and the overall shape of the polygon. In this article, we will explore the relationship between central angles and regular polygons, and provide solutions to several problems related to this topic.
Measuring Central Angles in Regular Polygons
A regular polygon is a polygon with equal sides and equal angles. The sum of the measures of the central angles in a polygon is always 360°. To find the measure of a central angle in a regular polygon, we can use the formula:
Measure of central angle = 360° / Number of sides
Problem 1: Which of the following measures of central angles yield a regular polygon?
A. 6° B. 14° C. 80° D. 40°
To solve this problem, we need to find the number of sides of a regular polygon that would yield a central angle of 6°, 14°, 80°, or 40°. We can use the formula:
Number of sides = 360° / Measure of central angle
Let's try each option:
- A. 6°: Number of sides = 360° / 6° = 60
- B. 14°: Number of sides = 360° / 14° = 25.71 (not a whole number)
- C. 80°: Number of sides = 360° / 80° = 4.5 (not a whole number)
- D. 40°: Number of sides = 360° / 40° = 9
Only option A yields a whole number of sides, which is 60. Therefore, the correct answer is A. 6°.
Problem 2: Find the number of sides of a regular polygon whose measure of the central angle is 12°.
To find the number of sides of a regular polygon with a central angle of 12°, we can use the formula:
Number of sides = 360° / Measure of central angle
Number of sides = 360° / 12° = 30
Therefore, the number of sides of a regular polygon with a central angle of 12° is 30.
Problem 3: Find the perimeter, area, and apothem of a regular polygon with 20 sides.
To find the perimeter, area, and apothem of a regular polygon with 20 sides, we need to know the length of one side. Let's call it 's'. The perimeter of a regular polygon is the sum of the lengths of all its sides. Since all sides are equal, the perimeter is:
Perimeter = Number of sides × Length of one side = 20 × s
The area of a regular polygon can be found using the formula:
Area = (Number of sides × s^2) / (4 × tan(π/Number of sides))
Area = (20 × s^2) / (4 × tan(π/20))
The apothem of a regular polygon is the distance from the center of the polygon to one of its sides. The apothem can be found using the formula:
Apothem = s / (2 × tan(π/Number of sides)) = s / (2 × tan(π/20))
Let's assume the length of one side is 10 units. Then:
Perimeter = 20 × 10 = 200 units Area = (20 × 10^2) / (4 × tan(π/20)) ≈ 346.41 square units Apothem = 10 / (2 × tan(π/20)) ≈ 5.24 units
Therefore, the perimeter, area, and apothem of a regular polygon with 20 sides and a side length of 10 units are 200 units, 346.41 square units, and 5.24 units, respectively.
Conclusion
Q: What is a central angle in a regular polygon?
A: A central angle is an angle whose vertex is at the center of a circle, and its sides pass through the endpoints of an arc. In a regular polygon, the central angle is the angle formed by two adjacent sides of the polygon.
Q: How do I find the measure of a central angle in a regular polygon?
A: To find the measure of a central angle in a regular polygon, you can use the formula: Measure of central angle = 360° / Number of sides.
Q: What is the relationship between the number of sides of a regular polygon and the measure of its central angle?
A: The number of sides of a regular polygon is inversely proportional to the measure of its central angle. As the number of sides increases, the measure of the central angle decreases.
Q: Can a regular polygon have a central angle of 90°?
A: No, a regular polygon cannot have a central angle of 90°. The sum of the measures of the central angles in a polygon is always 360°, and a 90° angle would require a polygon with only 4 sides, which is not a regular polygon.
Q: How do I find the perimeter of a regular polygon?
A: To find the perimeter of a regular polygon, you need to know the length of one side. The perimeter is then calculated by multiplying the number of sides by the length of one side.
Q: What is the apothem of a regular polygon?
A: The apothem of a regular polygon is the distance from the center of the polygon to one of its sides.
Q: How do I find the apothem of a regular polygon?
A: To find the apothem of a regular polygon, you can use the formula: Apothem = s / (2 × tan(π/Number of sides)), where 's' is the length of one side.
Q: Can a regular polygon have an apothem of 0 units?
A: No, a regular polygon cannot have an apothem of 0 units. The apothem is always greater than 0, as it is a distance from the center of the polygon to one of its sides.
Q: What is the relationship between the area of a regular polygon and the number of its sides?
A: The area of a regular polygon is directly proportional to the number of its sides. As the number of sides increases, the area of the polygon also increases.
Q: How do I find the area of a regular polygon?
A: To find the area of a regular polygon, you can use the formula: Area = (Number of sides × s^2) / (4 × tan(π/Number of sides)), where 's' is the length of one side.
Q: Can a regular polygon have an area of 0 square units?
A: No, a regular polygon cannot have an area of 0 square units. The area of a polygon is always greater than 0, as it is a measure of the space enclosed by the polygon.
Conclusion
In conclusion, central angles play a crucial role in determining the number of sides and the overall shape of a regular polygon. By understanding the relationship between central angles and regular polygons, you can solve problems related to the perimeter, area, and apothem of regular polygons.