Total Surface Area Of Spherical Sector Given Spherical Cap Height Calculator

Total Surface Area of Spherical Sector Formula:

\[ TSA = \pi \times r_{Sphere} \times \left( (2 \times h_{Cap}) + \sqrt{h_{Cap} \times ((2 \times r_{Sphere}) - h_{Cap})} \right) \]

Total Surface Area (TSA):

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1. What is Total Surface Area of Spherical Sector?

The Total Surface Area of a Spherical Sector refers to the entire area covering the curved surface of the spherical sector, including the spherical cap and the lateral surface. It is a measure of the total two-dimensional space enclosed on the entire surface.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \pi \times r_{Sphere} \times \left( (2 \times h_{Cap}) + \sqrt{h_{Cap} \times ((2 \times r_{Sphere}) - h_{Cap})} \right) \]

Where:

\( TSA \) — Total Surface Area of Spherical Sector (m²)
\( r_{Sphere} \) — Spherical Radius (m)
\( h_{Cap} \) — Spherical Cap Height (m)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula calculates the total surface area by considering both the cap height and the spherical radius, incorporating the square root function to account for the geometric properties.

3. Importance of TSA Calculation

Details: Calculating the total surface area is essential in various fields such as architecture, engineering, and physics, where understanding the surface properties of spherical sectors is required for design and analysis.

4. Using the Calculator

Tips: Enter the spherical radius and spherical cap height in meters. Ensure all values are positive and that the cap height does not 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 the apex at the center of the sphere, including a spherical cap.

Q2: Why is the square root function used in the formula?
A: The square root function helps in calculating the lateral surface area component based on the geometry of the spherical cap.

Q3: What are the units for TSA?
A: The total surface area is measured in square meters (m²), consistent with the input units for radius and height.

Q4: Can the cap height be greater than the spherical radius?
A: The cap height must be less than or equal to twice the spherical radius for the formula to be valid.

Q5: Is this formula applicable to all spherical sectors?
A: Yes, this formula is generally applicable to any spherical sector given the spherical radius and cap height.