Volume of Hollow Pyramid given Total Height and Missing Height Calculator

Volume of Hollow Pyramid Formula:

\[ V = \frac{\frac{1}{3} \times n \times (h_{Total} - h_{Missing}) \times l_{e(Base)}^2}{4 \times \tan(\frac{\pi}{n})} \]

Number of Base Vertices (n):

Total Height (h_Total):

Missing Height (h_Missing):

Edge Length of Base (l_e(Base)):

Volume of Hollow Pyramid (V):

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1. What is Volume of Hollow Pyramid?

The Volume of Hollow Pyramid is the total quantity of three-dimensional space enclosed by the surface of the Hollow Pyramid. It represents the volume difference between a complete pyramid and a removed smaller pyramid from its apex.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ V = \frac{\frac{1}{3} \times n \times (h_{Total} - h_{Missing}) \times l_{e(Base)}^2}{4 \times \tan(\frac{\pi}{n})} \]

Where:

\( V \) — Volume of Hollow Pyramid (m³)
\( n \) — Number of Base Vertices of Hollow Pyramid
\( h_{Total} \) — Total Height of Hollow Pyramid (m)
\( h_{Missing} \) — Missing Height of Hollow Pyramid (m)
\( l_{e(Base)} \) — Edge Length of Base of Hollow Pyramid (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \tan \) — Tangent trigonometric function

Explanation: The formula calculates the volume difference between a complete pyramid and a removed pyramid section, considering the polygonal base geometry.

3. Importance of Volume Calculation

Details: Accurate volume calculation is crucial for architectural design, structural engineering, material estimation, and various geometric applications involving hollow pyramid structures.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the minimum number of base vertices allowed?
A: The minimum number of base vertices is 3, which corresponds to a triangular base pyramid.

Q2: Can the missing height be zero?
A: Yes, if missing height is zero, the hollow pyramid becomes a complete solid pyramid.

Q3: What happens if missing height equals total height?
A: This would result in zero volume as the entire pyramid would be removed.

Q4: Are there any limitations to this formula?
A: The formula assumes regular polygonal bases and requires that the removed pyramid is similar to and concentric with the complete pyramid.

Q5: What units should be used for input values?
A: All linear measurements should be in consistent units (meters recommended), and the volume result will be in cubic units of the input.