Inradius of Heptagon given Short Diagonal Calculator

Inradius of Heptagon Formula:

\[ r_i = \frac{\frac{d_{Short}}{2 \times \cos(\frac{\pi}{7})}}{2 \times \tan(\frac{\pi}{7})} \]

Inradius of Heptagon:

1. What is Inradius of Heptagon given Short Diagonal?

Definition: The inradius of a heptagon is the radius of the circle inscribed within the heptagon that touches all its sides.

Purpose: This calculator determines the inradius when you know the length of the short diagonal of a regular heptagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{\frac{d_{Short}}{2 \times \cos(\frac{\pi}{7})}}{2 \times \tan(\frac{\pi}{7})} \]

Where:

\( r_i \) — Inradius of the heptagon
\( d_{Short} \) — Short diagonal of the heptagon (±5%)
\( \pi \) — Pi constant (approximately 3.14159)

Explanation: The formula first calculates the side length from the short diagonal, then uses trigonometric functions to determine the inradius.

3. Importance of Inradius Calculation

Details: Knowing the inradius is important for geometric constructions, area calculations, and understanding the proportions of regular heptagons.

4. Using the Calculator

Tips: Enter the short diagonal length (±5% tolerance). The value must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular heptagon?
A: A regular heptagon is a seven-sided polygon with all sides equal and all angles equal.

Q2: Why is there a ±5% range?
A: This accounts for measurement tolerances in practical applications.

Q3: How is the short diagonal defined?
A: The short diagonal connects two non-adjacent vertices skipping one vertex in between.

Q4: Can I use this for irregular heptagons?
A: No, this calculator only works for regular heptagons where all sides and angles are equal.

Q5: What are typical applications of this calculation?
A: Useful in architecture, engineering, and design involving seven-sided structures.