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GEOMETRY The area of a regular $n$ -sided polygon inscribed in a circle of radius 1 is given by$$A=\frac{n}{2} \sin \frac{360^{\circ}}{n}$$(A) Find $A$ for $n=8, n=100, n=1,000,$ and $n=10,000$ Compute each to five decimal places.(B) What number does $A$ seem to approach as $n \rightarrow \infty ?$ (What is the area of a circle with radius $1 ?$ )(GRAPH CANT COPY)

$=\pi r^{2}$$=\pi \times(1)^{2}$$=3.14159$

Precalculus

Algebra

Chapter 5

Trigonometric Functions

Section 4

Properties of Trigonometric Functions

Trigonometry

Functions

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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all right. We have a polygon inscribed in a circle, and we have a formula that calculates the area of that polygon where a is equal 10 divided by two times to sign of 3 63 160 degrees that is divided by N. And we know the radius of our circle is one. So we're going to calculate the area of this polygon that's inside this circle with a radius of one for several values. We're gonna start where N is equal to eight. So the area is going to be eight divided by two times the sign of 360 divided by AIDS. And when we calculate that we get approximately 2.8 to age for and then we do this for N is equal to 100 at 1000 will get the 1000 next. So area there's gonna be 100 over to times the sign of 360 over 100 and that's going to be approximately 3.13 953 and next we're going to do n is equal. 1000. We get a equals 1000 divided by two times the sign of 360 divided by 1000. And when we calculate this out, we should get approximately 3.1 for 157 No, I asked. We're going to calculate our area when n equals 10,000 and then is representing. By the way, I didn't mention this, but N is representing the number of sides in the polygon. Right? So we're increasing the number of sides we started with eight sides on 101,000. Now we want to know what the area of a polygon that has 10,000 sides is if it's inscribed in a circle, has a radius of one. Well, our area there's gonna be 10,000 divided by two times the sign of 360 divided by 10,000 in this is approximately gonna equal three points. 14159 Now, if we look and all of these values When we had eight sides, we had approximately 2.8. When we had 100 sides, we had approximately 3.13953 When we have 1000 signs, we have approximately 3.14157 10,000 signs. We have approximately 3.14159 So as and is increasing right is in his approaching infinity. It looks like our area is getting closer and closer to pie. Right? And if you think about the area of a circle whose radius is one, which is what this polygon is sitting inside, then area is gonna equal pi our square. That's gonna be pi times one squared. The area of the circle is pie, so this kind of makes sense.

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