Edge Length of Pentagon given Inradius using Interior Angle Calculator

Formula Used:

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

Edge Length of Pentagon:

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

The formula calculates the edge length of a regular pentagon when the inradius (radius of the inscribed circle) is known, using the interior angle properties of a pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( l_e \) — Edge Length of Pentagon (m)
\( r_i \) — Inradius 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 inradius, interior angles, and side lengths of a regular pentagon.

3. Importance of Edge Length Calculation

Details: Calculating the edge length from the inradius is essential in geometry, architecture, and engineering for designing and constructing pentagonal structures with specific inscribed circle properties.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and greater than zero for valid 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 degrees each).

Q2: What is the inradius of a pentagon?
A: The inradius is the radius of the circle that can be inscribed inside the pentagon, touching all five sides.

Q3: Why use trigonometric functions in this calculation?
A: Trigonometric functions help establish the relationship between the inradius 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, engineering projects, and geometric modeling where pentagonal shapes with specific inscribed circle properties are required.