Surface to Volume Ratio of Hollow Pyramid Calculator

Surface to Volume Ratio Formula:

\[ RA/V = \frac{n \cdot l_e(Base)/2 \cdot \left( \sqrt{h_{Total}^2 + \left( \frac{l_e(Base)^2}{4} \cdot \left( \cot\left( \frac{\pi}{n} \right) \right)^2 \right)} + \sqrt{h_{Missing}^2 + \left( \frac{l_e(Base)^2}{4} \cdot \left( \cot\left( \frac{\pi}{n} \right) \right)^2 \right)} \right)}{\frac{1}{3} \cdot n \cdot (h_{Total} - h_{Missing}) \cdot \frac{l_e(Base)^2}{4 \cdot \tan\left( \frac{\pi}{n} \right)}} \]

Number of Base Vertices (n):

Edge Length of Base (lₑ):

Total Height (h_Total):

Missing Height (h_Missing):

Surface to Volume Ratio (RA/V):

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1. What is Surface to Volume Ratio of Hollow Pyramid?

The Surface to Volume Ratio of Hollow Pyramid is the numerical ratio of the total surface area of the Hollow Pyramid to the volume of the Hollow Pyramid. It represents how much surface area is available per unit volume of the hollow pyramid structure.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

Where:

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

Explanation: The formula calculates the ratio by considering the surface area contributions from both the outer and inner surfaces relative to the volume of the hollow space.

3. Importance of Surface to Volume Ratio Calculation

Details: Surface to volume ratio is crucial in various engineering and architectural applications, particularly in heat transfer analysis, structural efficiency studies, and material optimization where the relationship between surface area and enclosed volume affects performance characteristics.

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. Ensure all values are positive and missing height is 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 while maintaining the external pyramid shape.

Q2: Why is surface to volume ratio important?
A: It indicates how efficiently a structure can exchange heat, chemicals, or other properties with its environment relative to its size.

Q3: What are typical values for this ratio?
A: The ratio varies significantly based on dimensions, but generally ranges from 0.5 to 5 m⁻¹ for most practical hollow pyramid structures.

Q4: Can this calculator handle different base shapes?
A: The calculator is specifically designed for regular polygonal bases where all sides and angles are equal.

Q5: What if the missing height equals total height?
A: This would represent a completely hollow structure with no volume, making the ratio undefined (division by zero).