Problem 7 - Algebra 2 and Trigonometry

Question

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 \(n .\)

Step-by-Step Solution

Answer

The perimeter formula is \(P = nr \times \tan \left(\frac{\pi}{n}\right)\) for a polygon with \(n\) sides.

Step 1: Measure of Angle AOB in Radians

Since square \(ABCD\) is symmetric about the circle, \(\triangle AOB\) is an isosceles right triangle. The angle \(\angle AOB\) is a right angle (90 degrees), which is equivalent to \(\frac{\pi}{2}\) radians.

Step 2: Finding Theta in Terms of AOB

\(\theta\) is half of \(\angle AOB\), so \(\theta = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}\).

Step 3: Expression for AP

\(AP\) is opposite \(\theta\) in right triangle \(\triangle AOP\). Thus, \(AP = r \times \tan \theta\).

Step 4: Expression for AB in Terms of tan(θ) and r

Since \(AP = r \times \tan \theta\) and the square's sides are twice this length, \(AB = 2 \times AP = 2r \times \tan \theta\).

Step 5: Perimeter of the Square

The perimeter of square \(ABCD\) is \(4 \times AB = 8r \times \tan \theta\). For a square, \(n=4\), so this simplifies to: \(4r \times n \times \tan \theta\).

Step 6: Repeat for Pentagon ABCDE

Using symmetry and a similar process, the central angle for a pentagon is \(\frac{2\pi}{5}\). So \(\theta = \frac{\pi}{5}\). Then, \(AB = 2r \times \tan \left(\frac{\pi}{10}\right)\), and the perimeter is \(5 \times 2r \times \tan \left(\frac{\pi}{10}\right) = 10r \times \tan \left(\frac{\pi}{10}\right)\).

Step 7: Identifying the Pattern

The expressions for the perimeters of both the square and the pentagon include terms \(n \times 2r \times \tan \left(\frac{\pi}{n}\right)\). This suggests a general form where the perimeter \(P = nr \times \tan \left(\frac{\pi}{n}\right)\) for a polygon with \(n\) sides circumscribed around the circle.

Key Concepts

Tangent Function

The tangent function is a key concept in trigonometry often denoted by \( \tan \theta \), where \( \theta \) is an angle in a right triangle. It is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to it. In the context of a circumscribed polygon, understanding the tangent function helps in calculating the lengths of the sides of polygons that touch or are 'tangent' to a circle.

Why is this important here? In a scenario where a regular polygon is circumscribed around a circle, each side of the polygon will just touch the circle at one point. This point is where the radius of the circle is perpendicular to the polygon's side.

Consider a polygon side \( AB \) being tangent to the circle at point \( P \). If we know \( \theta \), the angle subtended by the radius at the center of the circle, we can calculate \( AP \) (half of \( AB \) in the case of a square) using \( AP = r \times \tan \theta \). This relationship is essential for deriving the entire perimeter of the polygon.

Regular Polygon Perimeter

To find the perimeter of a regular, circumscribed polygon, we need to systematically account for the contribution of each side in the context of the circle around which it is drawn.

For a square, we discover that each side \( s \) is calculated as \( 2r \times \tan \theta \). When extended to all four sides, the perimeter \( P \) is simply \( 4 \times s \). For polygons with \( n \) sides, this expands to a general formula of \( P = n \times (2r \times \tan(\theta)) \), where the angle \( \theta \) depends on \( \frac{\pi}{n} \) reflecting the rotational symmetry of the polygon.

The pattern here reveals an elegant symmetry: The complete circumferential engagement of a polygon as it relates to a circle can invariably be summarized using the regular polygon's side interactions via the tangent function. This configuration is consistent across shapes like pentagons as well, depending solely on the circle's radius and the number of sides.

Circle Radius

The radius of a circle, denoted as \( r \), is the constant distance from the center of the circle to any point on its circumference. For a polygon circumscribed around this circle, the radius holds significant importance as each polygon side touches the circle precisely at the endpoints created by the radius. This geometric property ensures that all points of tangency on the polygon form a regular and symmetrical network.

When analyzing circumscribed polygons, the radius helps in framing expressions for side lengths and eventually the perimeter. Each side's length might be a factor of the radius, as in the formula \( s = 2r \times \tan \left( \frac{\pi}{n} \right) \), thereby suggesting that the radius is not only central to the geometric construction but also to calculating total external measures like perimeters.

This constancy and relation with other parameters like angle and sides underline the harmonious structure between the circle and the polygon it circumscribes, emphasizing circles' intrinsic impact on calculating regular polygon properties.