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Circumradius of Decagon Formula:
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The circumradius of a decagon is the radius of a circle that passes through all ten vertices of the decagon. It represents the distance from the center of the decagon to any of its vertices.
The calculator uses the circumradius formula:
Where:
Explanation: The formula calculates the circumradius based on the perimeter of the decagon, using the golden ratio relationship that appears in regular decagons.
Details: Calculating the circumradius is essential in geometry for determining the size and proportions of regular decagons, which is useful in architectural design, engineering applications, and mathematical problem-solving.
Tips: Enter the perimeter of the decagon in meters. The value must be positive and greater than zero. The calculator will automatically compute the circumradius.
Q1: What is a regular decagon?
A: A regular decagon is a ten-sided polygon where all sides are equal in length and all interior angles are equal (144 degrees each).
Q2: How is the circumradius related to the side length?
A: For a regular decagon with side length s, the circumradius is \( r_c = \frac{s}{2} \times (1+\sqrt{5}) \).
Q3: What is the significance of the golden ratio in this formula?
A: The term \( \frac{1+\sqrt{5}}{2} \) represents the golden ratio (φ ≈ 1.618), which appears frequently in the geometry of regular pentagons and decagons.
Q4: Can this formula be used for irregular decagons?
A: No, this formula applies only to regular decagons where all sides and angles are equal. Irregular decagons don't have a circumradius in the same sense.
Q5: How accurate is this calculation?
A: The calculation is mathematically exact for regular decagons, as it's derived from geometric principles using the golden ratio.