Circumradius of Pentagon given Edge Length using Interior Angle Calculator

Formula Used:

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

Circumradius of Pentagon:

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1. What is the Circumradius of Pentagon?

The Circumradius of Pentagon is the radius of a circumcircle touching each of the vertices of Pentagon. It's an important geometric property used in various mathematical and engineering applications.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: This formula calculates the circumradius using the edge length and trigonometric functions based on the interior angles of a regular pentagon.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is crucial for geometric constructions, architectural designs, and various engineering applications involving pentagonal shapes.

4. Using the Calculator

Tips: Enter the edge length of the pentagon in meters. The value must be positive and greater than zero.

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 degrees each).

Q2: How is the circumradius related to the edge length?
A: The circumradius is directly proportional to the edge length in a regular pentagon, with the relationship defined by trigonometric functions.

Q3: Can this calculator be used for irregular pentagons?
A: No, this calculator is specifically designed for regular pentagons where all sides and angles are equal.

Q4: What are practical applications of circumradius calculation?
A: Applications include architectural design, mechanical engineering, computer graphics, and geometric modeling.

Q5: How accurate is the calculation?
A: The calculation uses precise trigonometric functions and provides results accurate to 12 decimal places.