Circumradius Of Hendecagon Given Diagonal Across Five Sides Calculator

Formula Used:

\[ r_c = \frac{d_5}{2 \times \sin\left(\frac{5\pi}{11}\right)} \]

Circumradius Of Hendecagon (rₐ):

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1. What Is The Circumradius Of Hendecagon?

The circumradius of a hendecagon (11-sided polygon) is the radius of a circle that passes through all its vertices. It's an important geometric property that helps in understanding the polygon's size and proportions.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{d_5}{2 \times \sin\left(\frac{5\pi}{11}\right)} \]

Where:

\( r_c \) — Circumradius of hendecagon
\( d_5 \) — Diagonal across five sides of hendecagon
\( \pi \) — Mathematical constant (approximately 3.14159)
\( \sin \) — Trigonometric sine function

Explanation: This formula relates the circumradius to the diagonal across five sides using trigonometric relationships specific to the 11-sided polygon.

3. Importance Of Circumradius Calculation

Details: Calculating the circumradius is essential in geometry, architecture, and engineering for designing and analyzing regular polygons. It helps determine the polygon's overall size and its relationship to circumscribed circles.

4. Using The Calculator

Tips: Enter the diagonal across five sides in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a hendecagon?
A: A hendecagon is a polygon with 11 sides and 11 angles. When all sides and angles are equal, it's called a regular hendecagon.

Q2: Why use the diagonal across five sides specifically?
A: Different diagonals in a regular polygon have specific trigonometric relationships with the circumradius. The diagonal across five sides provides one such precise relationship for calculation.

Q3: Can this formula be used for irregular hendecagons?
A: No, this formula applies only to regular hendecagons where all sides and angles are equal.

Q4: What are practical applications of this calculation?
A: This calculation is used in architectural design, engineering projects involving polygonal structures, and in various mathematical and geometric analyses.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular hendecagons, with accuracy depending on the precision of the input value and the implementation of the trigonometric functions.