Area of Pentagon given Circumradius using Central Angle Calculator

Pentagon Area Formula:

\[ A = \frac{5 \times (r_c \times \sin(\pi/5))^2}{\tan(\pi/5)} \]

Area of Pentagon:

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1. What is the Pentagon Area Formula?

The formula calculates the area of a regular pentagon using its circumradius. A regular pentagon has all sides and angles equal, making this formula applicable when the distance from the center to any vertex (circumradius) is known.

2. How Does the Calculator Work?

The calculator uses the pentagon area formula:

\[ A = \frac{5 \times (r_c \times \sin(\pi/5))^2}{\tan(\pi/5)} \]

Where:

\( A \) — Area of the pentagon
\( r_c \) — Circumradius of the pentagon
\( \pi \) — Mathematical constant (approximately 3.14159)
\( \sin \) — Sine trigonometric function
\( \tan \) — Tangent trigonometric function

Explanation: The formula utilizes trigonometric functions to calculate the area based on the central angle (72°) of a regular pentagon.

3. Importance of Pentagon Area Calculation

Details: Calculating the area of a pentagon is essential in geometry, architecture, engineering, and various design applications where pentagonal shapes are used.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is a circumradius?
A: The circumradius is the radius of a circle that passes through all the vertices of the pentagon.

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

Q3: What are the units for the result?
A: The area result is in square meters (m²), matching the units of the input circumradius.

Q4: How accurate is this calculation?
A: The calculation uses precise mathematical constants and trigonometric functions, providing highly accurate results.

Q5: What if I know the side length instead of circumradius?
A: Different formulas are available for calculating pentagon area from side length. This calculator specifically uses circumradius.