Graphical Reasoning Consider an n -sided regular polygon...

Graphical Reasoning Consider an $n$ -sided regular polygon inscribed in a circle of radius $r .$ Join the vertices of the polygon to the center of the circle, forming $n$ congruent triangles (see figure). (a) Determine the central angle $\theta$ in terms of $n .$ (b) Show that the area of each triangle is $\frac{1}{2} r^{2} \sin \theta$ . (c) Let $A_{n}$ be the sum of the areas of the $n$ triangles. Find $\lim _{n \rightarrow \infty} A_{n} .$

Graphical Reasoning Consider an $n$ -sided regular polygon inscribed in a circle of radius $r .$ Join the vertices of the polygon to the center of the circle, forming $n$ congruent triangles (see figure).
(a) Determine the central angle $\theta$ in terms of $n .$
(b) Show that the area of each triangle is $\frac{1}{2} r^{2} \sin \theta$ .
(c) Let $A_{n}$ be the sum of the areas of the $n$ triangles. Find $\lim _{n \rightarrow \infty} A_{n} .$

Key Concepts

Regular Polygon

A regular polygon is a geometric figure with all sides and interior angles equal. When inscribed in a circle, its vertices are evenly spaced along the circumference, making it a fundamental example in understanding symmetry and its relationship with circular geometry.

Central Angle

The central angle is the angle formed at the center of the circle by lines drawn to two consecutive vertices of the polygon. It is determined by dividing the total angle around a point (360° or 2? radians) by the number of sides, n, ensuring that each of the n triangles formed is congruent.

Triangle Area Using the Sine Formula

The area of a triangle can be calculated using the formula (1/2)ab sin(C), where a and b are the lengths of two sides and C is the included angle. In the context of an inscribed regular polygon, these sides are radii of the circle and the included angle is the central angle, making the formula directly applicable to find the area of each triangle.

Limit and Convergence to the Circle's Area

As the number of sides n of an inscribed regular polygon increases, the polygon increasingly approximates the circle. The limit of the sum of the areas of the triangles (forming the polygon) as n approaches infinity is the area of the circle, illustrating a key concept in analysis and the idea of convergence in geometric approximations.

Transcript

00:01 So you have the following problem.

00:02 There's a, there's some pizza that you want to slice, want to divide into n equals slices.

00:21 So it's not a perfect triumph, but imagine that you have, imagine that that angle were the same for all of these pieces.

00:35 So i go there, theta.

00:37 And we have some radius here.

00:46 So that we do the whole turn, we get 2 pi.

00:58 And then each of these slices has an angle theta and well there's end of them.

01:04 End slices.

01:06 So you want to figure out, you have those end slices and the radius.

01:14 What is the area? what is the area of that shape? that area there.

01:22 Well, if you believe that each slice as the same area what's going to be in times a sub and where a suben is the area of a single slice here that shaded area there that's one of those pieces a suven so for a suben you have the following setting so there is a triangle here you have this angle theta and well this distance there is the same as a distance now there are isoselous triangle would well we can cut it in half and fold it to see that the total area here is going to be equal to well if you fold it it is going to be equal to the area of this rectangle right where all here we're here you have this angle that's theta halves and that is the same angle up there a half so that well if you do we have this triangle one of those of this lower triangle is the same as that upper triangle so here we have the house there are halves and here you have r that r here is that distance so well the area of these rectangle is going to be this distance distance x is going to be equal to the distance x here times the distance y right x times y that is the area of a sub n x times y but uh well what is what is x x is r cosine of theta halves r cause theta halves no that is a parent that is y r cause and uh that is y and uh that is y and x yeah my but and x is equal to r sign of theta hops so also that if you did the product you get the tell area is going to be r squared well sign of their hubs times sign of theta hops sign of theta hubs and well the double angle identities tell you that this gadget there that is equal to one one half of sign of that angle in there twice.

04:58 So one half of sign of this angle two times.

05:05 Two times.

05:08 So that wall, very nice thing happens here...

Please give Ace some feedback