Calculating the Side Length of a Regular Pentagon Inscribed in a Circle

Introduction to Regular Pentagons and Inscribed Circles

A regular pentagon is a five-sided polygon with equal sides and equal angles. When this pentagon is inscribed in a circle, each vertex of the pentagon touches the circumference of the circle. The problem at hand involves calculating the side length of a regular pentagon that is inscribed in a circle with a radius of 6 inches.

Methods for Calculating the Side Length

To find the length of the side of a regular pentagon inscribed in a circle, we can use several methods involving trigonometry and geometry. Let's explore these methods step by step.

Method 1: Trigonometric Formula

We can use the formula for the side length of a regular polygon inscribed in a circle:

[Formula: $s r cdot 2 cdot sinleft(frac{pi}{n}right)$]

where (n) is the number of sides and (r) is the radius of the circle. For a pentagon, (n 5) and the radius (r 6) inches.

Step 1: Substitute the values into the formula:

$s 6 cdot 2 cdot sinleft(frac{pi}{5}right)$

Step 2: Calculate (sinleft(frac{pi}{5}right)):

(sinleft(frac{pi}{5}right) approx 0.5878)

Step 3: Substitute the sine value into the formula:

(s 6 cdot 2 cdot 0.5878 approx 6 cdot 1.1756 approx 7.0536)

Thus, the length of each side of the regular pentagon is approximately 7.05 inches.

Method 2: Using Trigonometric Ratios in Right Triangles

Another approach is to use the properties of isosceles triangles and trigonometric ratios. Let's denote (s) as the side length of the pentagon. Let (O) be the center of the inscribed circle and (H) be the point where the radius meets the side of the pentagon. The angle at the center of the circle subtended by one side of the pentagon is:

2θ (2 × 5 - 4 × 90°) / 5 108°

θ 108° / 5 54°

We can form a right triangle with the radius (r 6) inches and the half-side length (s/2). The tangent of half the interior angle gives:

[Formula: $tan 54^circ frac{6}{frac{s}{2}}$]

Step 1: Rearrange the formula to solve for (s):

(s frac{12}{tan 54^circ} approx 8.71851)

Thus, the side length of the pentagon is approximately 8.71851 cm, or 8.71 inches.

Method 3: Isosceles Triangle Area

The pentagon can also be viewed as five isosceles triangles with a central angle of 72° and two equal sides of 6 inches. The length of one side of the pentagon can be calculated as:

(s 5 cdot sin(72^circ / 2) 5 cdot sin(36^circ) approx 2.94) inches

However, this method tends to result in a shorter side length, which may not be accurate for the given radius.

Conclusion

From the calculations above, the side length of a regular pentagon inscribed in a circle with a 6-inch radius is approximately 7.05 inches using the trigonometric formula. This method provides a more accurate result compared to other geometric approaches.

Additional Insights

The methods explored in this article demonstrate the versatility of trigonometry in solving geometric problems related to regular polygons inscribed in circles. Understanding these concepts can be beneficial in various fields, including architecture, engineering, and design.

Keywords: inscribed pentagon, side length, regular polygon, circle circumference, trigonometry