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Spherical Cap Radius of Spherical Sector Calculator

Formula Used:

\[ r_{Cap} = \sqrt{h_{Cap} \times ((2 \times r_{Sphere}) - h_{Cap})} \]

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1. What is the Spherical Cap Radius of Spherical Sector?

The Spherical Cap Radius of a Spherical Sector is defined as the distance between the center and any point on the circumference of the circle at the bottom level of the cap surface of the Spherical Sector. It is a crucial parameter in spherical geometry and 3D modeling.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Cap} = \sqrt{h_{Cap} \times ((2 \times r_{Sphere}) - h_{Cap})} \]

Where:

Explanation: This formula derives from the geometric properties of a sphere and its spherical cap, using the Pythagorean theorem in three dimensions.

3. Importance of Spherical Cap Radius Calculation

Details: Calculating the spherical cap radius is essential in various fields including architecture, astronomy, and engineering, particularly when dealing with spherical segments and volumes.

4. Using the Calculator

Tips: Enter the spherical cap height and spherical radius in meters. Both values must be positive, and the cap height cannot exceed twice the spherical radius.

5. Frequently Asked Questions (FAQ)

Q1: What is a spherical sector?
A: A spherical sector is a portion of a sphere defined by a conical boundary with apex at the sphere's center and the spherical cap as its base.

Q2: How is this different from a spherical cap?
A: A spherical cap is just the "cap" portion, while a spherical sector includes both the cap and the conical section connecting it to the sphere's center.

Q3: What are the units of measurement?
A: The calculator uses meters, but the formula works with any consistent unit of length (cm, mm, inches, etc.).

Q4: What if my cap height is greater than the sphere's diameter?
A: The formula is only valid when the cap height is between 0 and 2 times the spherical radius. Larger values are geometrically impossible.

Q5: Can this formula be used for hemispheres?
A: Yes, when the cap height equals the spherical radius, the cap radius equals the spherical radius, forming a hemisphere.

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