Inner Height of Hollow Pyramid given Volume Calculator

Formula Used:

\[ h_{Inner} = \frac{12 \times V \times \tan(\pi/n)}{n \times l_{e(Base)}^2} \]

Inner Height of Hollow Pyramid (h_Inner):

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1. What is the Inner Height of Hollow Pyramid?

The Inner Height of Hollow Pyramid is the length of the perpendicular from the apex of the complete pyramid to the apex of the removed pyramid in the Hollow Pyramid structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ h_{Inner} = \frac{12 \times V \times \tan(\pi/n)}{n \times l_{e(Base)}^2} \]

Where:

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

Explanation: This formula calculates the inner height based on the volume, number of base vertices, and edge length of the base, using trigonometric relationships in a regular polygonal pyramid structure.

3. Importance of Inner Height Calculation

Details: Calculating the inner height is crucial for understanding the geometric properties of hollow pyramids, which have applications in architecture, engineering design, and mathematical modeling of three-dimensional structures.

4. Using the Calculator

Tips: Enter the volume in cubic meters, number of base vertices (must be at least 3), and edge length of base in meters. All values must be 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 while maintaining the external pyramid shape.

Q2: Why is the tangent function used in this formula?
A: The tangent function relates the angle at the pyramid's apex to the ratio of height to half the base length, which is essential for calculating geometric properties of pyramids.

Q3: What is the minimum number of base vertices required?
A: The minimum number of base vertices is 3, which corresponds to a triangular base pyramid. The formula works for any regular polygonal base with 3 or more vertices.

Q4: Can this calculator handle different units?
A: The calculator uses meters for length units and cubic meters for volume. Convert other units to these standard metric units before calculation.

Q5: What are typical applications of this calculation?
A: This calculation is used in architectural design, structural engineering, 3D modeling, and mathematical analysis of pyramid-shaped structures with hollow interiors.