Inradius of Pentagon given Area using Interior Angle Calculator

Formula Used:

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

Inradius of Pentagon (rᵢ):

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1. What is the Inradius of Pentagon?

The inradius of a pentagon is the radius of the circle that fits perfectly inside the pentagon, touching all five sides. 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:

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

Where:

\( r_i \) — Inradius of the pentagon
\( A \) — Area of the pentagon
\( \pi \) — Mathematical constant (approximately 3.14159)
\( \cos \) — Cosine trigonometric function
\( \sin \) — Sine trigonometric function

Explanation: This formula calculates the inradius based on the area of a regular pentagon using trigonometric functions of the interior angles.

3. Importance of Inradius Calculation

Details: The inradius is important in geometry for determining the size of the inscribed circle, calculating various pentagon properties, and solving problems related to regular pentagons in engineering and design applications.

4. Using the Calculator

Tips: Enter the area of the pentagon in square 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 degrees each).

Q2: How is the inradius related to the circumradius?
A: The inradius is always smaller than the circumradius (radius of the circumscribed circle) in a regular pentagon.

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

Q4: What are practical applications of pentagon inradius?
A: Used in architecture, design, engineering, and various mathematical calculations involving regular pentagonal shapes.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular pentagons, using exact trigonometric values based on the interior angles.