Solving for the Side Length of a Regular Pentagon Given Its Area
In this article, we will explore how to find the side length of a regular pentagon given its area. The process involves utilizing the specific formula for the area of a regular polygon and some basic trigonometry. Let's delve into the step-by-step method and mathematical concepts required to find the solution.
Introduction to the Problem
A regular pentagon is a five-sided polygon with equal sides and angles. For a regular pentagon with an area of 338 square meters (m2), we aim to determine its side length from the given information. The formula for the area of a regular pentagon is: [ A frac{1}{4} sqrt{5 2sqrt{5}} s^2 ] Where (A) represents the area and (s) is the length of a side of the pentagon.
Step-by-Step Solution
Step 1: Set Up the Equation
Given that the area (A 338) m2, we can set up the equation as follows: [ 338 frac{1}{4} sqrt{5 2sqrt{5}} s^2 ]
Step 2: Isolate the Side Length Term
Multiply both sides of the equation by 4 to get rid of the fraction:
[ 1352 sqrt{5 2sqrt{5}} s^2 ]Now, we calculate the constant (sqrt{5 2sqrt{5}}), which is approximately 5.236:
[ sqrt{5 2sqrt{5}} approx 5.236 ]Step 3: Substitute and Solve for (s^2)
Substitute this value back into the equation:
[ 1352 5.236 s^2 ]Step 4: Isolate (s^2)
Divide both sides by 5.236 to solve for (s^2):
[ s^2 frac{1352}{5.236} approx 258.1 ]Step 5: Calculate the Side Length (s)
Take the square root of both sides to find (s):
[ s approx sqrt{258.1} approx 16.1 text{ meters} ]Therefore, the side length of the regular pentagon is approximately 16.1 meters.
Alternative Approach Using Triangle Geometry
An alternative method involves using the properties of the triangles formed within the regular pentagon. We know that each of the five central angles of a regular pentagon is 72°, and each of the internal angles is 108°. Each internal triangle, formed by a radius of the circumscribed circle and two sides of the pentagon, has a certain area which can be calculated.
Step 1: Calculate the Area of One Triangle
The area of one of these triangles is:
[ text{Area of one triangle} frac{338}{5} 67.6 text{ m}^2 ]Step 2: Use Trigonometry to Find the Base and Height
The triangle can be split into two right triangles with angles 36°, 54°, and 90°. Using the tangent function, we can find the relationship between the base (half the side length) and the height (apothem) of the triangle. The tangent of 36° gives us the ratio of the base to the height:
[ text{tan}(36°) frac{text{base}}{2 cdot text{height}} ]Given that the area of the triangle is 67.6 m2, we can equate the area to (frac{1}{2} cdot text{base} cdot text{height}):
[ frac{1}{2} cdot 2 cdot text{height} cdot text{tan}(36°) 67.6 ]Let's solve for (text{height}):
[ text{height} cdot text{tan}(36°) 67.6 ]Since (text{tan}(36°) approx 0.7265), we have:
[ text{height} cdot 0.7265 67.6 ]Solving for (text{height}):
[ text{height} frac{67.6}{0.7265} approx 93.3 text{ m} ]Now, calculate the base:
[ text{base} 2 cdot text{height} cdot text{tan}(36°) 2 cdot 93.3 cdot 0.7265 approx 134.6 text{ m} ]Finally, the side length of the pentagon is twice the base:
[ text{side length} 2 cdot 134.6 approx 269.2 text{ meters} ]Conclusion
The side length of a regular pentagon with an area of 338 square meters, using the initial method, is approximately 16.1 meters. An alternative approach using triangle properties and trigonometry yields a different result, emphasizing the importance of method selection and calculation accuracy in geometry.
For related geometric problems, such as calculating areas and side lengths of other regular polygons, this article provides a valuable insight into the underlying mathematical principles.