Volume of Pentagon Calculator

Volume Formulas:

For pentagonal prism: \( V = \frac{5}{4} \tan(54°) \times s^2 \times h \)

For pentagonal pyramid: \( V = \frac{1}{3} \times \frac{5}{4} \tan(54°) \times s^2 \times h \)

Volume (V):

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1. What is the Volume of Pentagon?

The volume of a pentagonal shape (prism or pyramid) is the space it occupies in three dimensions. A pentagonal prism has two parallel pentagonal bases connected by rectangular faces, while a pentagonal pyramid has a pentagonal base and triangular faces meeting at a common vertex (apex).

2. How Does the Calculator Work?

The calculator uses these formulas:

For pentagonal prism: \( V = \frac{5}{4} \tan(54°) \times s^2 \times h \)

For pentagonal pyramid: \( V = \frac{1}{3} \times \frac{5}{4} \tan(54°) \times s^2 \times h \)

Where:

\( V \) — Volume in cubic meters (m³)
\( s \) — Side length of the pentagon in meters (m)
\( h \) — Height of the prism or pyramid in meters (m)
\( \tan(54°) \) — Mathematical constant for regular pentagon calculation

Explanation: The formula first calculates the area of the pentagonal base, then multiplies by height (for prism) or 1/3 of height (for pyramid).

3. Importance of Volume Calculation

Details: Calculating volume is essential in architecture, engineering, packaging, and any field dealing with three-dimensional space. It helps determine capacity, material requirements, and structural properties.

4. Using the Calculator

Tips: Enter the side length of the pentagon and the height of the shape. All values must be positive numbers. Select whether you're calculating for a prism or pyramid.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a pentagonal prism and pyramid?
A: A prism has two parallel pentagonal bases connected by rectangular faces, while a pyramid has one pentagonal base and triangular faces meeting at an apex.

Q2: Why is tan(54°) used in the formula?
A: This trigonometric function helps calculate the area of a regular pentagon based on its side length.

Q3: What units should I use?
A: The calculator uses meters, but the formula works with any consistent unit (cm, inches, etc.) as long as all measurements use the same unit.

Q4: Does this work for irregular pentagons?
A: No, these formulas are only for regular pentagons (all sides and angles equal).

Q5: How accurate is the calculation?
A: The calculation is mathematically precise for perfect regular pentagonal shapes.