Recall from your geometry course that a polygon is...

Recall from your geometry course that a polygon is circumscribed about a circle if each side of the polygon is tangent to the circle. Since each side is tangent to the circle, the radius of the circle is perpendicular to each side at the point of tangency. We will use the tangent tunction to examine the formula for the perimeter of a circumscribed regular polygon. Let square $A B C D$ be circumscribed about circle $O .$ A radius of the circle, $\overline{O P},$ is perpendicular to $\overline{A B}$ at $P .$ (1) In radians, what is the measure of $\angle A O B ?$ (2) Let $m \angle A O P=\theta .$ If $\theta$ is equal to one-half the measure of $\angle A O B$ , find $\theta .$ (3) Write an expression for $A P$ in terms of $\tan \theta$ and $r,$ the radius of the circle. (4) Write an expression for $A B=s$ in terms of $\tan \theta$ and $r$ (5) Use part $(4)$ to write an expression for the perimeter in terms of $r$ and the number of sides, $n$ . b. Let regular pentagon $A B C D E$ be circumscribed about circle $O .$ Repeat part a using pentagon $A B C D E .$ c. Do you see a pattern in the formulas for the perimeter of the square and of the pentagon? If so, make a conjecture for the formula for the perimeter of a circumscribed regular polygon in terms of the radius $r$ and the number of sides $n .$

Recall from your geometry course that a polygon is circumscribed about a circle if each side of the polygon is tangent to the circle. Since each side is tangent to the circle, the radius of the circle is perpendicular to each side at the point of tangency. We will use the tangent tunction to examine the formula for the perimeter of a circumscribed regular polygon.
Let square $A B C D$ be circumscribed about circle $O .$ A radius of the circle, $\overline{O P},$ is perpendicular to $\overline{A B}$ at $P .$
(1) In radians, what is the measure of $\angle A O B ?$
(2) Let $m \angle A O P=\theta .$ If $\theta$ is equal to one-half the measure of $\angle A O B$ , find $\theta .$
(3) Write an expression for $A P$ in terms of $\tan \theta$ and $r,$ the radius of the circle.
(4) Write an expression for $A B=s$ in terms of $\tan \theta$ and $r$
(5) Use part $(4)$ to write an expression for the perimeter in terms of $r$ and the number of sides, $n$ .
b. Let regular pentagon $A B C D E$ be circumscribed about circle $O .$ Repeat part a using pentagon $A B C D E .$
c. Do you see a pattern in the formulas for the perimeter of the square and of the pentagon? If so, make a conjecture for the formula for the perimeter of a circumscribed regular polygon in terms of the radius $r$ and the number of sides $n .$

Key Concepts

Perimeter Calculation of Regular Polygons

The perimeter of a regular polygon is calculated by multiplying the length of one side by the number of sides. When the polygon is circumscribed about a circle, the side length can be expressed in terms of the radius and the tangent of a specific angle derived from the central angle. This relationship provides a generalized method to compute the perimeter based on the circle’s radius and the number of sides of the polygon.

Circumscribed Regular Polygon

A circumscribed regular polygon is one that surrounds a circle such that every side of the polygon is tangent to the circle. This configuration implies that the circle touches each side exactly once, and the polygon’s symmetry ensures that the points of tangency are evenly distributed around the circle.

Tangent Line and Perpendicular Radius

When a line is tangent to a circle, the radius drawn to the point of tangency is perpendicular to the tangent line. This fundamental geometric property is crucial in forming right triangles, which can be analyzed using trigonometric functions to relate the dimensions of the circle and the polygon.

Central Angle and Its Division

In a regular polygon, the central angle is the angle subtended at the center of the circle by one side of the polygon. For a polygon with n sides, the full angle around the center (2? radians) is divided equally, making each central angle 2?/n radians. Dividing this angle further, for instance into two equal parts, is often useful for applying trigonometric ratios in associated right triangles.

Trigonometric Relationships in Right Triangles

In the right triangles formed by the radius, the line from the center to a vertex, and the line to the point of tangency, trigonometric functions such as the tangent function become useful. Specifically, the tangent of half the central angle relates the length of the side of the polygon (or segment thereof) to the radius of the circle, thereby linking angular measures with linear dimensions.

Transcript

00:01 We will begin with a square that is circumscribed about a circle.

00:10 Each side of the square is length s.

00:15 The radius of the circle, o .p, is perpendicular to a side of the square.

00:23 And so for part a, number one, we want the angle a -o -b, the measure of angle a -o -b, the measure of angle a -o -b, in radiance and actually we can see that angle aob is actually one -fourth because there are four equal angles around that circle.

00:54 So area, the measure of angle a -o -b is actually one -fourth of two -pie, which is a full circle, or that would be pi over four, which is pi over two, 2 radians for the measure of angle aob.

01:14 For part 2, they want the measure of angle aop, which is actually half the measure of angle aob.

01:25 So it's half of pi over 2, which makes it pi over 4 radians.

01:35 For 3, we want the length of ap in terms of r and the tam.

01:43 Tangent of theta.

01:45 We'll notice that the tangent of theta is equal to the opposite leg, which is length ap, over the adjacent leg, which is the length r.

02:01 So to get ap, we have to do, multiply both sides by r.

02:07 We'll get r times the tangent of theta, in this case theta is pi over four, is equal to the length of ap.

02:19 For part four, we want the length ab in terms of r and tangent of theta.

02:30 Ab is actually twice the length of ap, so it will be two times r times r, the tangent of pi over four if we do the substitution.

02:47 And for five, we want the perimeter...

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