Total Surface Area Of Hollow Pyramid Given Inner Height And Total Height Calculator

Formula Used:

\[ TSA = n \times \frac{le_{Base}}{2} \times \left( \sqrt{h_{Total}^2 + \left( \frac{le_{Base}^2}{4} \times \left( \cot\left( \frac{\pi}{n} \right) \right)^2 \right)} + \sqrt{(h_{Total} - h_{Inner})^2 + \left( \frac{le_{Base}^2}{4} \times \left( \cot\left( \frac{\pi}{n} \right) \right)^2 \right)} \right) \]

Number of Base Vertices (n):

Edge Length of Base (le_Base):

Total Height (h_Total):

Inner Height (h_Inner):

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 pyramid, including both the outer and inner surfaces. It represents the complete surface area that would be visible if the pyramid were unfolded.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

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

3. Importance of TSA Calculation

Details: Calculating the total surface area is crucial for various applications including material estimation, construction planning, structural analysis, and cost calculation for hollow pyramid structures.

4. Using the Calculator

Tips: Enter the number of base vertices (minimum 3), edge length of base, total height, and inner height. All values must be positive, and inner height must be less than total height.

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 inside.

Q2: Why is the number of base vertices important?
A: The number of vertices determines the shape of the base (triangle, square, pentagon, etc.) and affects the surface area calculation through trigonometric functions.

Q3: What units should I use for measurements?
A: Use consistent units (preferably meters) for all length measurements. The result will be in square units of the input measurement.

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.

Q5: What if the inner height equals the total height?
A: This would mean no hollow space exists, and the calculator would treat it as a solid pyramid.