Area of a Circle as a Limit

1. Use the Step button to increase the number of sides, given by n, from the default value 3 to 16. Which of the following best describes what happens to the polygon on the screen?

(a)As n increases, both the length of the individual sides and the total polygon length increase.
(b)As n increases, the length of the individual sides decreases while the total polygon length remains constant.
(c)As n increases, the length of the individual sides decreases while the total polygon length increases.
(d)As n increases, both the length of the individual sides and the total polygon length decrease.

2. Let capital L sub capital P of n

L_{P}(n)

be the total length of the polygon with n sides. Suppose we want to use capital L sub capital P of n

L_{P}(n)

to approximate the circumference of the circle. What is the smallest value of n for which capital L sub capital P of n

L_{P}(n)

is within nought point o five

0.05

of the circumference of the circle?

(a)7
(b)15
(c)21
(d)30

3. Which of the following is correct for all n?

(a)capital L sub capital P of two n equals two multiplied by capital L sub capital P of n $L_{P}(2n) = 2L_{P}(n)$
(b)capital L sub capital P of n is less than capital L sub capital P of two n is less than two multiplied by pi $L_{P}(n) < L_{P}(2n) < 2 \pi$
(c)two multiplied by pi is less than capital L sub capital P of two n is less than capital L sub capital P of n $2 \pi < L_{P}(2n) < L_{P}(n)$
(d)capital L sub capital P of sum of n and one equals capital L sub capital P of n plus one $L_{P}(n+1) < L_{P}(n)+1$

PREVIEW: AREA OF A CIRCLE AS LIMIT OF THE AREA OF A POLYGON

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