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Let An be the area of a regular polygon with n sides. It is given that A1000 is approximately equal to 314 cm2. Also, the vertices of this 1000 sided polygon are numbered clockwise in order, from 1 to 1000. What is the approximate distance between the vertices 1 and 501?
- a)15.7 cm
- b)18.1 cm
- c)20 cm
- d)21.6 cm
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let An be the area of a regular polygon with n sides. It is given that...
Solution: It can be seen that, the higher the number of sides of a regular polygon, the more closely does its area approach to that of its circum-circle.
In this case, we have a polygon of 1000 sides and its area will be very close to that of the circle of radius r.
To find r, we put, πr2= 314 cm2
In this case, we have a polygon of 1000 sides and its area will be very close to that of the circle of radius r.
To find r, we put, πr2= 314 cm2
So r~ 10 cm
Now, vertices 1 and 501 of our 1000 sided polygon will correspond to the opposite ends of the diameter of the circum-circle of this polygon. The distance between them will be approximately = 2 * r = 20 cm Hence, option 3.
Now, vertices 1 and 501 of our 1000 sided polygon will correspond to the opposite ends of the diameter of the circum-circle of this polygon. The distance between them will be approximately = 2 * r = 20 cm Hence, option 3.
Most Upvoted Answer
Let An be the area of a regular polygon with n sides. It is given that...
To find the approximate distance between the vertices 1 and 501 of a regular polygon with 1000 sides, we can use the concept of the circumradius of a regular polygon.
Concept: Circumradius of a Regular Polygon
The circumradius of a regular polygon is the distance between the center of the polygon and any of its vertices. It is denoted by R.
Formula: Circumradius of a Regular Polygon
The formula to calculate the circumradius of a regular polygon is:
R = s / (2 * sin(π/n))
where s is the length of each side of the polygon and n is the number of sides.
Approach:
1. We are given that the area of the polygon is approximately equal to 314 cm². Let's assume the side length of the polygon is 'a'.
2. The formula to calculate the area of a regular polygon is:
Area = (n * a²) / (4 * tan(π/n))
3. Substituting the given values, we get:
314 = (1000 * a²) / (4 * tan(π/1000))
4. Simplifying the equation, we find:
a² = (4 * 314 * tan(π/1000)) / 1000
5. Taking the square root of both sides, we get:
a = √((4 * 314 * tan(π/1000)) / 1000)
6. Now, we can use the formula for the circumradius to find the distance between vertices 1 and 501:
R = a / (2 * sin(π/1000))
7. Substituting the value of 'a' calculated in step 5, we can find the value of R.
8. Finally, the distance between vertices 1 and 501 is given by 2R, as the polygon is regular and has 1000 sides.
Calculation:
Using the above approach, we can calculate the value of R and the distance between vertices 1 and 501.
R = (√((4 * 314 * tan(π/1000)) / 1000)) / (2 * sin(π/1000))
R ≈ 9.98 cm
Distance between vertices 1 and 501 = 2R ≈ 2 * 9.98 ≈ 19.96 cm ≈ 20 cm
Hence, the approximate distance between the vertices 1 and 501 is 20 cm, which corresponds to option (c).
Concept: Circumradius of a Regular Polygon
The circumradius of a regular polygon is the distance between the center of the polygon and any of its vertices. It is denoted by R.
Formula: Circumradius of a Regular Polygon
The formula to calculate the circumradius of a regular polygon is:
R = s / (2 * sin(π/n))
where s is the length of each side of the polygon and n is the number of sides.
Approach:
1. We are given that the area of the polygon is approximately equal to 314 cm². Let's assume the side length of the polygon is 'a'.
2. The formula to calculate the area of a regular polygon is:
Area = (n * a²) / (4 * tan(π/n))
3. Substituting the given values, we get:
314 = (1000 * a²) / (4 * tan(π/1000))
4. Simplifying the equation, we find:
a² = (4 * 314 * tan(π/1000)) / 1000
5. Taking the square root of both sides, we get:
a = √((4 * 314 * tan(π/1000)) / 1000)
6. Now, we can use the formula for the circumradius to find the distance between vertices 1 and 501:
R = a / (2 * sin(π/1000))
7. Substituting the value of 'a' calculated in step 5, we can find the value of R.
8. Finally, the distance between vertices 1 and 501 is given by 2R, as the polygon is regular and has 1000 sides.
Calculation:
Using the above approach, we can calculate the value of R and the distance between vertices 1 and 501.
R = (√((4 * 314 * tan(π/1000)) / 1000)) / (2 * sin(π/1000))
R ≈ 9.98 cm
Distance between vertices 1 and 501 = 2R ≈ 2 * 9.98 ≈ 19.96 cm ≈ 20 cm
Hence, the approximate distance between the vertices 1 and 501 is 20 cm, which corresponds to option (c).
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Let An be the area of a regular polygon with n sides. It is given that A1000 is approximately equal to 314 cm2. Also, the vertices of this 1000 sided polygon are numbered clockwise in order, from 1 to 1000. What is the approximate distance between the vertices 1 and 501?a)15.7 cmb)18.1 cmc)20 cmd)21.6 cmCorrect answer is option 'C'. Can you explain this answer?
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