Edge Length of Pentagon given Circumradius using Interior Angle Calculator

Formula Used:

\[ l_e = r_c \times \frac{\sin\left(\frac{3}{5}\pi\right)}{\frac{1}{2} - \cos\left(\frac{3}{5}\pi\right)} \]

Edge Length of Pentagon:

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1. What is the Edge Length of Pentagon given Circumradius using Interior Angle?

This formula calculates the edge length of a regular pentagon when the circumradius (radius of the circumscribed circle) is known, using the interior angle relationships of a pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l_e = r_c \times \frac{\sin\left(\frac{3}{5}\pi\right)}{\frac{1}{2} - \cos\left(\frac{3}{5}\pi\right)} \]

Where:

\( l_e \) — Edge Length of Pentagon (m)
\( r_c \) — Circumradius of Pentagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \cos \) — Cosine trigonometric function

Explanation: The formula derives from the geometric relationships between the circumradius and the sides of a regular pentagon, utilizing trigonometric functions with the interior angle of 108° (3π/5 radians).

3. Importance of Edge Length Calculation

Details: Calculating the edge length from circumradius is essential in geometric design, architecture, and engineering applications involving pentagonal shapes. It helps in determining the precise dimensions of pentagonal structures.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular pentagon?
A: A regular pentagon is a five-sided polygon where all sides are equal in length and all interior angles are equal (108° each).

Q2: What is circumradius?
A: Circumradius is the radius of a circle that passes through all the vertices (corner points) of a polygon.

Q3: Why use trigonometric functions in this calculation?
A: Trigonometric functions help establish the relationship between the circumradius and the side length through the interior angles of the pentagon.

Q4: Can this formula be used for irregular pentagons?
A: No, this formula applies only to regular pentagons where all sides and angles are equal.

Q5: What are practical applications of this calculation?
A: This calculation is used in architectural design, construction planning, manufacturing of pentagonal components, and various engineering applications.