Inradius of Pentagon given Circumradius using Interior Angle Calculator

Inradius of Pentagon given Circumradius using Interior Angle Formula:

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

Inradius of Pentagon (r_i):

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1. What is Inradius of Pentagon given Circumradius using Interior Angle?

The Inradius of Pentagon given Circumradius using Interior Angle is a geometric calculation that determines the radius of the circle inscribed inside a regular pentagon based on its circumradius and interior angle properties.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_i \) — Inradius of Pentagon (m)
\( r_c \) — Circumradius of Pentagon (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \cos \) — Cosine function

Explanation: This formula calculates the inradius of a regular pentagon using trigonometric relationships based on the circumradius and the interior angle properties of the pentagon.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry for determining various properties of pentagons, including area calculations, construction planning, and understanding the spatial relationships between inscribed and circumscribed circles.

4. Using the Calculator

Tips: Enter the circumradius 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° each).

Q2: What is the difference between inradius and circumradius?
A: The inradius is the radius of the circle inscribed inside the pentagon (touching all sides), while the circumradius is the radius of the circle that passes through all vertices of the pentagon.

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

Q4: What are practical applications of this calculation?
A: This calculation is used in architecture, engineering design, geometric modeling, and various mathematical applications involving pentagonal shapes.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular pentagons, with accuracy limited only by the precision of the input values and computational floating-point arithmetic.