Total Surface Area of Hollow Pyramid Calculator

Total Surface Area Formula:

\[ TSA = \frac{n \cdot l_e(Base)}{2} \cdot \left( \sqrt{h_{Total}^2 + \left( \frac{l_e(Base)^2}{4} \cdot \cot\left( \frac{\pi}{n} \right)^2 \right)} + \sqrt{h_{Missing}^2 + \left( \frac{l_e(Base)^2}{4} \cdot \cot\left( \frac{\pi}{n} \right)^2 \right)} \right) \]

Number of Base Vertices (n):

Edge Length of Base (le(Base)):

Total Height (hTotal):

Missing Height (hMissing):

Total Surface Area (TSA):

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1. What is Total Surface Area of Hollow Pyramid?

The Total Surface Area of a Hollow Pyramid is the total amount of two-dimensional space occupied by all the faces of the Hollow Pyramid, including both the outer and inner surfaces.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( n \) — Number of base vertices
\( l_e(Base) \) — Edge length of base (m)
\( h_{Total} \) — Total height of pyramid (m)
\( h_{Missing} \) — Missing height of hollow portion (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \cot \) — Cotangent trigonometric function

Explanation: The formula calculates the sum of the lateral surface areas of both the complete pyramid and the removed pyramid portion.

3. Importance of TSA Calculation

Details: Calculating the total surface area is crucial for determining material requirements, cost estimation, and structural analysis in architectural and engineering applications involving hollow pyramid structures.

4. Using the Calculator

Tips: Enter the number of base vertices (minimum 3), edge length of base in meters, total height in meters, and missing height in meters. All values must be valid positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a hollow pyramid?
A: A hollow pyramid is a pyramid structure with a smaller pyramid removed from its interior, creating a hollow space.

Q2: Why is the cotangent function used in the formula?
A: The cotangent function is used to calculate the apothem of the base polygon, which is essential for determining the slant heights of the pyramid faces.

Q3: What units should be used for input values?
A: All length measurements should be in consistent units (typically meters). The result will be in square units of the input.

Q4: Can this calculator handle different base shapes?
A: Yes, the formula works for any regular polygon base by adjusting the number of vertices parameter (n).

Q5: What if the missing height is zero?
A: If missing height is zero, the calculator will compute the surface area of a solid pyramid without any hollow portion.