Inscribe a regular n -sided polygon inside a circle of radius 1 and compute the area of the poly

step 1 All right, so this is section 5 .1, problem 21. So we're given a circle of radius r equals 1, an

Inscribe a regular n -sided polygon inside a circle of...

Key Concepts

Central Angle

The central angle is the angle subtended at the center of the circle by two consecutive vertices of the polygon. It is critical in forming the isosceles triangles that make up the polygon, and its measure (usually 2?/n for an n-sided polygon) is used in determining the area via trigonometric functions.

Area of a Polygon Using Trigonometry

The area of an inscribed regular polygon can be computed by dividing it into congruent isosceles triangles. The area of each triangle can be expressed as ½ R² sin(?), where R is the circle’s radius and ? is the central angle. The summation of these triangle areas yields the area of the entire polygon.

Regular Polygon

A regular polygon is a polygon with all sides and all angles equal. This uniformity simplifies many geometric calculations, such as computing the area, where symmetry plays a crucial role.

Inscribed Polygon

An inscribed polygon is a polygon whose vertices all lie on a circle. This configuration allows the use of the circle’s radius and central angles from the circle to compute properties like the side lengths and area of the polygon using trigonometric relationships.

Approaching the Area of a Circle

As the number of sides of an inscribed polygon increases, the polygon becomes a better approximation of the circle. The area of such a polygon approaches the area of the circle. This concept demonstrates how polygons can be used to approximate curved shapes and is foundational in understanding limits and the development of integral calculus.

Transcript

00:01 All right, so this is section 5 .1, problem 21.

00:06 So we're given a circle of radius r equals 1, and we have to fit a polygon with n size inside that circle.

00:14 So you can see there's a polygon with four sides, a square, and a polygon with eight sides, an octagon, and a polygon of 16th sides.

00:22 So we want to find the area of each of these polygons, and then we want to see how they relate to the area of the circle.

00:29 So for the first one, you can see there's a circle of radius of.

00:33 With a square inside of it.

00:35 I turned it diagonally in this case.

00:39 So we could find the area by multiplying the base times the height, but i think it would be a lot easier if we just find the area of this triangle right here and then multiply it by four, since there's four identical triangles inside this square.

00:55 I'll draw them now.

01:01 So the area equals four times the area of a triangle, which is one half, base times height.

01:17 So the base and height, i'll just put parentheses here.

01:24 Let's find what they are.

01:27 So as we can see, this side right here is the same as the radius.

01:31 So it's one.

01:32 And so is this side.

01:33 They're both one.

01:34 So this is the base.

01:35 This is the height.

01:37 So let's put ones inside our parentheses here.

01:41 So it ends up being four times one half, which is two.