Circumradius of Heptagon given Inradius Calculator

Formula Used:

\[ r_c = \frac{r_i \cdot \tan(\pi/7)}{\sin(\pi/7)} \]

Circumradius of Heptagon (r_c):

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1. What is the Circumradius of Heptagon given Inradius Formula?

The formula calculates the circumradius of a regular heptagon (7-sided polygon) when its inradius is known. It establishes the relationship between these two important geometric properties of a regular polygon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{r_i \cdot \tan(\pi/7)}{\sin(\pi/7)} \]

Where:

\( r_c \) — Circumradius of Heptagon (m)
\( r_i \) — Inradius of Heptagon (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \tan \) — Tangent trigonometric function
\( \sin \) — Sine trigonometric function

Explanation: The formula derives from the geometric relationships in a regular heptagon, using trigonometric functions to connect the inradius and circumradius through the central angle (π/7 radians).

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential for various geometric applications, including polygon construction, area calculations, and understanding the spatial properties of regular heptagons in design and engineering contexts.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and non-zero. The calculator will compute the corresponding circumradius using the trigonometric relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular heptagon?
A: A regular heptagon is a seven-sided polygon where all sides are equal in length and all interior angles are equal (approximately 128.57 degrees each).

Q2: What is the difference between circumradius and inradius?
A: The circumradius is the radius of the circle that passes through all vertices of the polygon, while the inradius is the radius of the circle inscribed within the polygon that touches all sides.

Q3: Can this formula be used for other polygons?
A: This specific formula applies only to heptagons. Other regular polygons have different formulas based on their number of sides.

Q4: What are practical applications of this calculation?
A: This calculation is used in architecture, engineering design, computer graphics, and any field requiring precise geometric constructions of seven-sided figures.

Q5: How accurate is the calculation?
A: The calculation uses precise trigonometric functions and provides high accuracy, limited only by the precision of the input value and computational floating-point arithmetic.