Total Surface Area of Spherical Sector Given Volume and Spherical Cap Height Calculator

Formula Used:

\[ TSA = \pi \times \sqrt{\frac{3 \times V}{2 \times \pi \times h_{Cap}}} \times (2 \times h_{Cap} + r_{Cap}) \]

Volume of Spherical Sector (V):

m³

Spherical Cap Height (h_Cap):

Spherical Cap Radius (r_Cap):

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 is the sum of the areas of the spherical cap and the lateral surface area of the conical portion. It represents the complete surface area of the three-dimensional shape formed by a portion of a sphere.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ TSA = \pi \times \sqrt{\frac{3 \times V}{2 \times \pi \times h_{Cap}}} \times (2 \times h_{Cap} + r_{Cap}) \]

Where:

\( TSA \) — Total Surface Area of Spherical Sector (m²)
\( V \) — Volume of Spherical Sector (m³)
\( h_{Cap} \) — Spherical Cap Height (m)
\( r_{Cap} \) — Spherical Cap Radius (m)
\( \pi \) — Mathematical constant pi (≈3.1416)

Explanation: This formula calculates the total surface area by incorporating the volume and dimensions of the spherical cap to determine the complete surface measurement.

3. Importance of TSA Calculation

Details: Calculating the total surface area of a spherical sector is important in various engineering, architectural, and manufacturing applications where precise surface measurements are needed for material estimation, heat transfer calculations, or structural design.

4. Using the Calculator

Tips: Enter the volume in cubic meters, spherical cap height in meters, and spherical cap radius in meters. All values must be positive numbers (radius can be zero for a complete hemisphere).

5. Frequently Asked Questions (FAQ)

Q1: What is a spherical sector?
A: A spherical sector is a solid portion of a sphere bounded by a spherical cap and the cone whose apex is the center of the sphere and whose base is the base of the cap.

Q2: Can the spherical cap radius be zero?
A: Yes, when the spherical cap radius is zero, the shape becomes a complete hemisphere.

Q3: What are the units of measurement?
A: The calculator uses meters for length dimensions, square meters for area, and cubic meters for volume. Consistent units must be used for accurate results.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the provided formula, assuming precise input values.

Q5: What if I get an error message?
A: Ensure all input values are positive numbers and that the volume, height, and radius values are consistent with a valid spherical sector.