Formula for the Circumference of a Circle (Grades 11-12)

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Formula for the Circumference of a Circle

Let circle A have a diameter of length 1.
Circle with Center Point A
1. 
If a regular n-sided polygon is inscribed in circle A, and then lines are drawn from the center of the circle to each vertex of the n-sided polygon, n congruent triangles are formed. What type of triangle are they, regardless of the value of n?
  1. Equilateral
  2. Isosceles
  3. Right
  4. Scalene
2. 
For one of the triangles described in the previous question, what is the length of the sides which touch the center of the circle?
  1. n/2
  2. 1
  3. 1/2
  4. 1/n
3. 
Let triangle ABC be one of the triangles of the inscribed polygon referred to in the previous questions. Here, A would be the center of the circle, B and C would lie on the circle, and [math]bar{BA}[/math] and [math]bar{AC}[/math] have lengths determined in the previous question. To determine the length of [math]bar{BC}[/math], which will be called [math]a[/math], one can use the formula [math]a=2r sin((360°)/(2n))[/math], where r is the radius of the circle and n is the number of sides of the inscribed polygon. Which of the following gives the best reasoning for why the argument of the sine function, which is half of [math]m ang BAC[/math], is [math](360°)/(2n) ?[/math]
Equilateral Triangle Height v2
  1. Since a triangle has 180°, to divide by 2n you must multiply the numerator by 2 as well.
  2. To account for the conversion from radians to degrees, 1/n is multiplied by 360°/2.
  3. It depends on the size of the circle, but in this case, where r = 1/2, 360°/n is multiplied by 1/2.
  4. The inscribed polygon is divided into n triangles, and then half of that would mean dividing a full rotation by 2n.
4. 
Using the formula from the previous question, find the perimeter of an inscribed regular hexagon in circle A.
  1. 3
  2. 1/2
  3. 6
  4. 12
5. 
Using the formula in part C, find the perimeter of an inscribed regular 20-sided polygon in circle A.
  1. 0.1564
  2. 3.1287
  3. 6.2574
  4. 0.3129
6. 
Using the formula in part C, find the perimeter of an inscribed regular 100-sided polygon in circle A.
  1. 30.9017
  2. 6.2822
  3. 0.0314
  4. 3.1411
7. 
Using the formula in part C, find the perimeter of an inscribed regular 200-sided polygon in circle A.
  1. 62.5738
  2. 6.2829
  3. 0.0157
  4. 3.1415
8. 
As the number of sides continues to increase, will the value of the perimeter continue to increase without bound? Or is it approaching some value? Why?
  1. The perimeter value will continue to increase without bound, because the number of sides will keep increasing.
  2. The perimeter value will continue to increase without bound, because n is increasing at a faster rate than a is decreasing.
  3. The perimeter value will approach a certain value, since it is bounded by the circumference of the circle. The value seems to be [math]pi[/math].
  4. The perimeter value will approach a certain value, because as n increases, a decreases. The value seems to be 4.
9. 
Thus far, only a circle of radius 1/2 was used. Which of the following formulas would correctly relate this circle with a new circle of unknown circumference, C, and radius, r. Why?
  1. [math]C/pi = r/(1/2)[/math] because all circles are similar.
  2. [math]C=1/2 r pi[/math] since the new circle will depend on the measurements of the circle already investigated.
  3. [math]C/2 = pi r^2[/math] because the area of a circle is related to its circumference.
  4. [math]C = npi, r=n/2[/math] since the new circle's circumference and radius will depend upon the number of sides of an inscribed regular polygon.
10. 
Rearranging the equation from the previous step, we find that [math]C = 2 pi r[/math], which is the formula for the circumference of a circle. If we had started with a circle such that [math]d!=1[/math], would we have been able to derive the correct formula? Why or why not?
  1. No, a circle with a larger or smaller diameter would have had a larger or smaller circumference, and so the value found in step H would have been different.
  2. No, a circle with [math]d!=1[/math] would also have [math]r!=1/2[/math], and so the formula would be different.
  3. Yes, because other calculations would be performed to compensate for this, and thus arrive at the correct formula.
  4. Yes, since the ratio of a circle's diameter to its circumference is always the same, it doesn't matter what the length of the original circle's diameter is.

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