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 that helps in understanding the spatial relationships within a regular pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

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

Explanation: This formula establishes the relationship between the circumradius and inradius of a regular pentagon using trigonometric functions based on the interior angles.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential in geometry, architecture, and engineering applications involving pentagonal shapes. It helps determine the size of the smallest circle that can contain the pentagon.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumradius and inradius?
A: In a regular pentagon, the circumradius is always larger than the inradius, and their ratio is constant for all regular pentagons.

Q2: Can this formula be used for irregular pentagons?
A: No, this formula specifically applies to regular pentagons where all sides and angles are equal.

Q3: What is the significance of the 3/5π term?
A: This term represents three-fifths of π radians, which relates to the interior angles of a regular pentagon (108° each).

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for regular pentagons, though computational precision may introduce minor rounding errors.

Q5: What are practical applications of this calculation?
A: This calculation is used in architectural design, mechanical engineering, and geometric modeling where pentagonal shapes are involved.