Circumradius Of Pentagon Given Area Using Interior Angle Calculator

Circumradius of Pentagon Formula:

\[ r_c = \sqrt{\frac{2 \times A}{5 \times \sin\left(\frac{3}{5}\pi\right)}} \]

Circumradius of Pentagon (r_c):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Circumradius of Pentagon?

The circumradius of a pentagon is the radius of the circumscribed circle that passes through all five vertices of the pentagon. It represents the distance from the center of the pentagon to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ r_c = \sqrt{\frac{2 \times A}{5 \times \sin\left(\frac{3}{5}\pi\right)}} \]

Where:

\( r_c \) — Circumradius of pentagon
\( A \) — Area of pentagon
\( \pi \) — Mathematical constant pi (approximately 3.14159)
\( \sin \) — Trigonometric sine function

Explanation: This formula calculates the circumradius based on the area of a regular pentagon and the interior angle properties of a pentagon.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry, architecture, and engineering applications where pentagonal shapes are used. It helps determine the size and proportions of pentagonal structures and designs.

4. Using the Calculator

Tips: Enter the area of the pentagon in square meters. The area must be a positive value 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: How is this formula derived?
A: The formula is derived from the relationship between the area of a regular pentagon and its circumradius, using trigonometric functions based on the pentagon's interior angles.

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

Q4: What are practical applications of pentagon circumradius?
A: Applications include architectural design (pentagonal buildings), engineering (pentagonal components), and various mathematical and geometric calculations.

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 limitations.