1. What is the Circumradius of Octagon given Height?

The circumradius of an octagon is the radius of the circumscribed circle that passes through all vertices of the octagon. This calculator determines the circumradius based on the height of a regular octagon.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( r_c \) — Circumradius of Octagon (m)
• \( h \) — Height of Octagon (m)

Explanation: The formula calculates the circumradius by multiplying the height of the octagon by the constant \(\sqrt{1 - \frac{1}{\sqrt{2}}}\), which is derived from the geometric properties of a regular octagon.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is essential in geometry and engineering for designing octagonal structures, determining spatial relationships, and solving geometric problems involving regular octagons.

4. Using the Calculator

Tips: Enter the height of the octagon in meters. The value must be positive and greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular octagon?
A: A regular octagon is an eight-sided polygon where all sides are equal in length and all interior angles are equal (135 degrees each).

Q2: How is height defined for an octagon?
A: The height of a regular octagon is the vertical distance between its top and bottom edges when oriented with one side horizontal.

Q3: What is the relationship between circumradius and height?
A: The circumradius is proportional to the height, with the constant factor \(\sqrt{1 - \frac{1}{\sqrt{2}}}\) ≈ 0.541196.

Q4: Can this formula be used for irregular octagons?
A: No, this formula applies only to regular octagons where all sides and angles are equal.

Q5: What are practical applications of this calculation?
A: This calculation is used in architecture, engineering design, manufacturing, and various geometric applications involving octagonal shapes.