Inradius of Pentagon Given Edge Length Using Interior Angle Calculator

Inradius of Pentagon Formula:

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

Inradius of Pentagon:

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1. What is the Inradius of Pentagon?

The Inradius of Pentagon is defined as the radius of the circle which is inscribed inside the Pentagon. It represents the distance from the center of the pentagon to any of its sides.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_i \) — Inradius 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 inradius using the edge length and trigonometric functions based on the interior angles of a regular pentagon.

3. Importance of Inradius Calculation

Details: The inradius is important in geometry for determining the size of the largest circle that can fit inside a pentagon, and is used in various engineering and design 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: Why use trigonometric functions in this calculation?
A: Trigonometric functions help relate the edge length to the inradius through the interior angles of the pentagon.

Q3: What are practical applications of this calculation?
A: This calculation is used in architecture, engineering design, and various mathematical problems involving pentagonal shapes.

Q4: How accurate is this formula?
A: The formula is mathematically exact for regular pentagons and provides precise results.

Q5: Can this calculator handle different units?
A: The calculator uses meters as the default unit, but you can convert other units to meters before input.