Volume of a Pentagonal Pyramid Calculator with Diameter

Pentagonal Pyramid Volume Formula:

\[ V = \frac{5}{12} \tan(54^\circ) \left(\frac{d}{2 \tan(36^\circ)}\right)^2 h \]

Volume (V):

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1. What is a Pentagonal Pyramid?

A pentagonal pyramid is a pyramid with a pentagonal base and five triangular faces that meet at a point (the apex). The volume represents the space enclosed within this three-dimensional shape.

2. How Does the Calculator Work?

The calculator uses the pentagonal pyramid volume formula:

\[ V = \frac{5}{12} \tan(54^\circ) \left(\frac{d}{2 \tan(36^\circ)}\right)^2 h \]

Where:

\( V \) — Volume in cubic meters
\( d \) — Diameter of the circumscribed circle around the pentagonal base
\( h \) — Height of the pyramid (perpendicular distance from base to apex)
\( \tan(54^\circ) \) — Trigonometric function of 54 degrees
\( \tan(36^\circ) \) — Trigonometric function of 36 degrees

Explanation: The formula first converts the diameter to the side length of the pentagon, then calculates the base area, and finally multiplies by height and 1/3 (incorporated in the 5/12 factor).

3. Importance of Volume Calculation

Details: Calculating the volume of a pentagonal pyramid is essential in architecture, geometry, and engineering applications where this shape is used in structures or design elements.

4. Using the Calculator

Tips: Enter the diameter of the circumscribed circle around the pentagonal base and the pyramid height. Both values must be positive numbers in meters.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between diameter and side length?
A: The diameter is for the circumscribed circle around the pentagon, not the length of the pentagon's sides. The calculator converts diameter to side length internally.

Q2: How accurate is this calculation?
A: The calculation is mathematically exact for a perfect regular pentagonal pyramid. Real-world measurements may introduce small errors.

Q3: Can I use this for irregular pentagonal pyramids?
A: No, this formula only works for regular pentagonal pyramids where the base is a perfect regular pentagon.

Q4: What are typical applications of this shape?
A: Pentagonal pyramids appear in architectural designs, certain crystal structures, and some decorative elements.

Q5: Why are the angles 54° and 36° used?
A: These angles come from the geometry of a regular pentagon (108° interior angle), with 54° being half of that and 36° being the complementary angle.