The perimeter of a regular polygon circumscribed about a...

The perimeter of a regular polygon circumscribed about a circle of radius $r$ is given by the formula $P=2 n r \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$. where $n$ represents the number of sides. (a) Rewrite the formula in terms of a single trig function; (b) verify the formula for a square circumscribed about a circle with radius $4 \mathrm{m} ;$ and (c) Find the perimeter of a dodecagon (12 sides) circumscribed about the same circle.

The perimeter of a regular polygon circumscribed about a circle of radius $r$ is given by the formula $P=2 n r \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$.
where $n$ represents the number of sides. (a) Rewrite the formula in terms of a single trig function;
(b) verify the formula for a square circumscribed about a circle with radius $4 \mathrm{m} ;$ and (c) Find the perimeter of a dodecagon (12 sides) circumscribed about the same circle.

Key Concepts

Application of Geometric Formulas

Applying general geometric and trigonometric formulas to specific cases reinforces understanding. It involves substituting given values, such as the number of sides or radius, into derived formulas, checking the validity of the formulas, and computing specific measurements like the perimeter.

Circumscribed Figures

A circumscribed figure is one where a polygon is drawn around a circle such that each side of the polygon is tangent to the circle. This geometric concept is key when relating the circle’s radius to the dimensions of the polygon, particularly in calculating the polygon's perimeter.

Regular Polygon Geometry

Regular polygons are figures with all sides and angles equal. Understanding the geometric properties of regular polygons, including their relationships with circumscribed or inscribed circles, is essential for deriving formulas related to perimeter, area, and side lengths.

Trigonometric Identities

Trigonometric identities, such as the one that expresses sin(?)/cos(?) as tan(?), are fundamental tools in mathematics. They allow for the simplification and rewriting of expressions, making it easier to manipulate and solve problems involving trigonometric functions.

Transcript

00:01 All right, in this problem, we're talking about the perimeter of a polygon.

00:04 And they've given us this equation here, 2nr times this quantity here.

00:10 And i've highlighted it because the first task they want us to do is to rewrite it as a trig function.

00:22 That's only one trig function.

00:24 Right now we have sine and cosine, but we only want one.

00:26 So, sine over cosine, of the same angle, we can rewrite.

00:31 As tangent.

00:34 So this is 2nr times tangent of pi over n.

00:41 The next task requires us to make sure that we are verifying this with a known value.

00:54 So in this case this is a primitive or polygon that's circumscribed around a circle of radius r.

01:01 That's where r comes in as the radius of that circle and then n is the number of sides for that polygon so in this case they want us to do a square and the square has four sides and then we also know that if r is the radius of the circumscribed figure we have something that looks like this a little bad drawing but that's okay we have r here that means that the side length of the square is going to be 2 r okay so 2r is the side length and we know that the perimeter of a square we know that that's going to be side plus side plus side or 4 s in this case so the perimeter of this is going to be 2 times times r.

02:10 But r we know is 4.

02:12 So i'm going to take this 4 out or r out and make it 4.

02:15 So it's 8 times 4 or in this case it's going to be 32.

02:20 So that's the answer that i'm going to expect from b by using this equation in a.

02:26 So the perimeter is going to be two times the number of sides, which we're dealing with a square.

02:33 The radius is going to be 4.

02:36 And then we're going to multiply this by tangent of pi over four...