The area of a regular polygon that has been circumscribed...

The area of a regular polygon that has been circumscribed about a circle of radius $r(\text { see figure })$ is given by the formula $A=n r^{2} \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$, CAN'T COPY THE GRAPH 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 area of a dodecagon ( 12 sides) circumscribed about the same circle.

The area of a regular polygon that has been circumscribed about a circle of radius $r(\text { see figure })$ is given by the formula $A=n r^{2} \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$,
CAN'T COPY THE GRAPH
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 area of a dodecagon ( 12 sides) circumscribed about the same circle.

Key Concepts

Regular Polygons

Regular polygons are figures with equal-length sides and equal angles. Their symmetry lends itself to using trigonometry to determine properties like area and perimeter, particularly when the polygon is related to an inscribed or circumscribed circle.

Circumscribed Figures

A circumscribed polygon is one that surrounds a circle such that each side of the polygon is tangent to the circle. This configuration establishes a relationship between the side length of the polygon and the radius of the circle, allowing the use of trigonometric functions to compute its properties.

Trigonometric Functions

Sine, cosine, and tangent functions are essential in connecting angular measures to the ratios of side lengths in right triangles. In this context, the identity tan(?) = sin(?)/cos(?) simplifies the formula, enabling the use of a single trigonometric function to express the area.

Area Calculation Using Trigonometry

The area of a regular polygon circumscribed about a circle can be derived by partitioning the figure into isosceles or right triangles and applying trigonometric ratios. This method not only facilitates algebraic manipulation but also allows for easy verification of area formulas for specific cases such as a square or a dodecagon.

Transcript

00:02 All right, in this problem, they're talking about a polygon that has a circle inscribed in it.

00:09 And i just want to, the only thing i want to add is talking about what n and r are.

00:15 N is the number of sides, and r is the radius of the inscribed circle.

00:26 All right, so the first part asks us to rewrite it so that we have only one trig function, and that's kind of actually easy.

00:33 I'm going to rewrite this and recognize that sign of something over cosine of that same something is actually a fancy way of saying tangent of that something.

00:44 So now we've simplified that.

00:47 The second part says, well, do it with a square that has a radius of four meters.

00:55 So a circle is being inscribed into a square.

00:58 The radius of that circle is four meters.

01:04 So let's just talk about the area of this square.

01:12 We know the area of squares.

01:18 The area of the square is side squared, right? so if this case it tells us that the radius is 4, if r equals 4, then that means the side is the entire diameter, right? so it's going to be 8.

01:38 So in this case, 8 squared is going to be 64.

01:41 So that's the answer that we're expecting to get.

01:44 Now let's verify this because a square is a polygon.

01:48 Let's verify it using this new formula, which is a little extra, but it works for polygons with more than four sides.

01:58 So here we have four sided polygon.

02:01 The radius is four.

02:04 That's going to be squared.

02:05 And then we're going to be tangent of pylosite.

02:08 Pi over four all right all ends become four and the cool thing here is that tangent of pi over four is just one and then this is actually four cubed or 16 times four which is 64 so that kind of agrees with what we expected to see with a square what they're asking in the last part is they want us to use a 12 -sided object that uses the same circle of radius 4...

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