# 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.

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?

- Equilateral
- Isosceles
- Right
- 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?

- n/2
- 1
- 1/2
- 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]

- Since a triangle has 180°, to divide by 2n you must multiply the numerator by 2 as well.
- To account for the conversion from radians to degrees, 1/n is multiplied by 360°/2.
- It depends on the size of the circle, but in this case, where r = 1/2, 360°/n is multiplied by 1/2.
- 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.

- 3
- 1/2
- 6
- 12

5.

Using the formula in part C, find the perimeter of an inscribed regular 20-sided polygon in circle A.

- 0.1564
- 3.1287
- 6.2574
- 0.3129

6.

Using the formula in part C, find the perimeter of an inscribed regular 100-sided polygon in circle A.

- 30.9017
- 6.2822
- 0.0314
- 3.1411

7.

Using the formula in part C, find the perimeter of an inscribed regular 200-sided polygon in circle A.

- 62.5738
- 6.2829
- 0.0157
- 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?

- The perimeter value will continue to increase without bound, because the number of sides will keep increasing.
- The perimeter value will continue to increase without bound, because n is increasing at a faster rate than a is decreasing.
- 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].
- 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?

- [math]C/pi = r/(1/2)[/math] because all circles are similar.
- [math]C=1/2 r pi[/math] since the new circle will depend on the measurements of the circle already investigated.
- [math]C/2 = pi r^2[/math] because the area of a circle is related to its circumference.
- [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?

- 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.
- No, a circle with [math]d!=1[/math] would also have [math]r!=1/2[/math], and so the formula would be different.
- Yes, because other calculations would be performed to compensate for this, and thus arrive at the correct formula.
- 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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