A regular polygon with n sides is inscribed in a circle with radius 1 . a. Explain why m ∠A O X=

A regular polygon with n sides is inscribed in a circle with...

Key Concepts

Area of a Regular Polygon

The area of a regular polygon is efficiently computed by dividing it into congruent isosceles triangles with a common vertex at the circle's center. The area of each triangle is found using trigonometric functions (sine and cosine), and summing these areas yields a formula for the entire polygon’s area.

Regular Polygon

A regular polygon is a convex figure with all sides of equal length and all interior angles equal. This uniformity means that many geometric properties are symmetric, allowing one to derive general formulas relating side length, perimeter, and area, especially when the polygon is inscribed in a circle.

Inscribed Polygon in a Circle

When a polygon is inscribed in a circle, each vertex of the polygon lies on the circle. This configuration creates a direct relationship between the polygon and the circle, making it possible to use central angles and chord properties to determine the side lengths and other measures related to the polygon.

Central Angles of Regular Polygons

The central angle of a regular polygon is the angle formed at the circle's center by two radii drawn to adjacent vertices, and it is found by dividing 360° by the number of sides. This central angle is crucial for establishing the measures of the constituent angles and for deriving trigonometric relationships within the polygon.

Chord Length and Trigonometry in Circles

Each side of an inscribed regular polygon acts as a chord of the circle, whose length can be expressed via trigonometric functions such as sine. Specifically, the chord length is related to the sine of half the central angle, linking the circle’s geometry with the polygon’s side.

Right Triangles in Circular Geometry

By drawing radii to the polygon’s vertices and constructing perpendiculars to the sides, right triangles are formed. These triangles allow the use of sine and cosine ratios to relate the angle measures to distances such as side lengths and distances from the center, forming the basis for expressing perimeter and area formulas.

Perimeter of a Regular Polygon

The perimeter of a regular polygon is the total length around it, obtained by multiplying the length of one side (found via chord relationships) by the number of sides. The derivation harnesses trigonometric identities to express the side lengths in terms of sine functions of the central angles.

Transcript

00:01 So the question gives us a regular polygon with n sides inscribed inside a circle with radius 1.

00:07 So the best way to do this is just to draw the circle.

00:11 This is going to help us imagine what we're talking about.

00:14 So this is just going to be the side.

00:17 So this is going to be a side here.

00:20 So this is a rough sketch.

00:21 So there's going to be however many sides n in total.

00:26 So these points here touching the circle are going to be points a.

00:32 Point b, so these will make a triangle by touching o in the middle and we'll do a dash line to help us with the equations later on.

00:43 With the midpoint, this line is denoted as x.

00:46 So we want to know why this angle here is going to equal 180 over n.

00:55 The best way to do this is we can easily work out this angle here.

01:01 So a to b simply going to be 360 degrees that are in a circle divided by n number of sides.

01:10 So a to b is going to be one side.

01:15 And there's sides going off in either direction, which are going to make more triangles.

01:21 So however many sides are going to be however many triangles.

01:24 So the number of that will just be denoted as n.

01:26 So aox is exactly half, as we can see by that dash line to the right angle, aob, that's just over 2.

01:39 We'll sub that top bit in, so 360 degrees over n divided by 2, scroll down, going to equal 180 degrees over n.

01:52 So that's the answer to part a.

01:54 I'll just put part a there.

01:58 So b must have to show that a x, so the length of a x here is equal to sign 180 over n.

02:08 So it's going to be easier if we just draw the triangle out.

02:11 So i'll just draw it on its side to make it easier to show.

02:16 So this here is going to be point o and this will be the right angle.

02:24 This will be x and this will be point a.

02:27 This is going to be r equals one.

02:31 So the best way we can work out the length of a of x is we can simply use the sign rule.

02:37 So we know that sine of x over r is going to equal the equivalent of sign of a of x equal equal sign of o.

02:54 Over a of x so we can ignore this middle bit just focus on those last two as a rule equivalent and we'll rearrange for a of x so a of x you're going to equal sign of o over sign of x times r and we know oh so this is the angle a over x which is also just the same as angle o here.

03:30 That's an o not a zero.

03:33 So we can plug that in and we know that r equals 1 from the diagram...

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