Problem 92 - Precalculus

Question

In calculus, some applications of the derivative require the solution of triangles. Solve each triangle using the Law of Cosines. A regular pentagon is inscribed in a circle of radius \(10 \mathrm{ft}\) Find its perimeter. Round your answer to the nearest tenth.

Step-by-Step Solution

Answer

The perimeter of the pentagon is approximately 59 ft.

Step 1: Understand the Pentagon Inscription

A regular pentagon inscribed in a circle with radius 10 ft means that each vertex of the pentagon lies on the circle. This circle is known as the circumcircle of the pentagon. Since it is regular, all sides of the pentagon are of equal length, and all interior angles are equal.

Step 2: Calculate the Central Angle

A regular pentagon has 5 equal sides, hence it subtends 5 equal central angles at the center of the circle. The total sum of angles in a circle is 360 degrees. Therefore, each central angle in the pentagon is calculated as:\[ \text{Central angle} = \frac{360}{5} = 72^\circ \]

Step 3: Apply the Law of Cosines

The Law of Cosines states that in a triangle with sides \( a, b, c \) and angle \( C \) opposite to side \( c \):\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]In the pentagon's triangle (formed by two radius and a side of the pentagon), we let the sides \( a \) and \( b \) be the radii (10 ft each) and \( C \) be the central angle (72°). Hence, the length of one side \( s \) of the pentagon becomes:\[ s^2 = 10^2 + 10^2 - 2 \cdot 10 \cdot 10 \cdot \cos(72^\circ) \]

Step 4: Solve for the Pentagon's Side Length

Substitute known values into the formula derived from the Law of Cosines:\[ s^2 = 100 + 100 - 200 \cos(72^\circ) \]Calculate \( \cos(72^\circ) \) and simplify:\[ \cos(72^\circ) \approx 0.309 \]\[ s^2 = 200 - 200 \times 0.309 \]\[ s^2 = 200 - 61.8 = 138.2 \]\[ s \approx \sqrt{138.2} \approx 11.8 \text{ ft} \]

Step 5: Calculate the Perimeter

The perimeter of the pentagon is simply 5 times the length of one side:\[ \text{Perimeter} = 5 \times 11.8 = 59 \text{ ft} \]Thus, the perimeter of the pentagon is approximately 59 ft.

Key Concepts

Pentagon

A pentagon is a five-sided polygon. It is called regular when all its sides and angles are the same. When a regular pentagon is mentioned, each interior angle measures \( 108^\circ \). Regular pentagons make calculations simpler compared to irregular ones because of their symmetry. The regular pentagon is often found in geometry problems due to its unique features, like symmetry and equidistance from its center to any vertex. This makes it perfect for inscription in a circle or creating a circumcircle.

Circumcircle

A circumcircle is a circle that passes through all vertices of a polygon. When you inscribe a regular pentagon in a circumcircle, this circle aids in connecting geometric ideas. Given any regular pentagon, its circumcircle simplifies solving for side lengths and angles because it provides a consistent radius. With the regular pentagon, you know any line from the pentagon's center to a vertex is a radius. This helps when applying formulas like the Law of Cosines, which was crucial in solving the original exercise by providing constant reference points.

Central Angle

The central angle is crucial for calculations with polygons like the pentagon when inscribed in a circle. It is the angle formed at the circle's center by radii stretching to adjacent vertices of the polygon. For a regular pentagon, this angle ensures equal division of the circle. Here's how it’s calculated:

• The entire circle's angles total 360 degrees.
• Divide by the number of sides (5 for a pentagon).
• Hence, each central angle is \( \frac{360}{5} = 72^\circ \).

Understanding this angle helps in using the Law of Cosines effectively to find side lengths of the pentagon.

Perimeter

The perimeter of a shape is the total length around it. For a regular pentagon, this is straightforward. Multiply the side length by the number of sides. In the context of the given pentagon inscribed in a circumcircle of radius 10 ft, once you find the side length using the Law of Cosines, the perimeter is just:

• Calculate one side, \( s \approx 11.8 \text{ ft} \).
• Perimeter is \( 5 \times s \).
• So, \( 5 \times 11.8 = 59 \text{ ft} \).

This direct multiplication method is due to the symmetry of the regular pentagon, making it simpler to calculate than irregular polygons.