Circumradius of Hexagon Given Inradius Calculator

Circumradius of Hexagon Formula:

\[ r_c = \frac{2 \times r_i}{\sqrt{3}} \]

Circumradius of Hexagon (r꜀):

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1. What is Circumradius of Hexagon Given Inradius?

The circumradius of a regular hexagon is the radius of the circle that passes through all its vertices. Given the inradius (the radius of the inscribed circle), the circumradius can be calculated using a specific mathematical relationship.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{2 \times r_i}{\sqrt{3}} \]

Where:

\( r_c \) — Circumradius of Hexagon (m)
\( r_i \) — Inradius of Hexagon (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)

Explanation: This formula derives from the geometric properties of a regular hexagon, where the relationship between the circumradius and inradius is fixed and can be expressed using the square root of 3.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in various geometric and engineering applications, including designing hexagonal structures, tessellations, and understanding the spatial properties of hexagons in both 2D and 3D contexts.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding circumradius using the mathematical relationship.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between circumradius and inradius?
A: The circumradius is the radius of the circle passing through all vertices of the hexagon, while the inradius is the radius of the circle inscribed within the hexagon, touching all sides.

Q2: Can this formula be used for irregular hexagons?
A: No, this formula applies only to regular hexagons where all sides and angles are equal. Irregular hexagons do not have a consistent relationship between circumradius and inradius.

Q3: What are practical applications of this calculation?
A: This calculation is useful in architecture, engineering design, material science (honeycomb structures), and various mathematical problems involving regular hexagons.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact for regular hexagons. The accuracy depends on the precision of the input value and the computational precision of the calculator.

Q5: Can I calculate inradius from circumradius using the same formula?
A: Yes, the formula can be rearranged as \( r_i = \frac{r_c \times \sqrt{3}}{2} \) to calculate inradius from circumradius.