Logo

MonoCalc

/

Pentagonal Prism Volume Calculator

Geometry

Pentagon Type

Side Length (a)

Height (h)

Unit

Decimal Places

ah

Volume

430.1194 cm³

Base Area

43.0119 cm²

Volume Conversions

mm³430119.3501
cm³430.1194
m³0.0004
in³26.2475
ft³0.0152

About This Tool

Pentagonal Prism Volume Calculator

A pentagonal prism is a three-dimensional solid with two parallel congruent pentagons as its bases and five rectangular lateral faces connecting them. This calculator computes the volume of any pentagonal prism — whether the base is a regular pentagon (all sides equal) or a custom pentagon where only the base area is known.

Volume Formula

The volume of any prism equals the area of its base multiplied by its height:

V = A × h

For a regular pentagonal prism, the base area is calculated from the side length a using:

A = (√(25 + 10√5) / 4) × a² ≈ 1.7205 × a²

So the full formula becomes:

V = (√(25 + 10√5) / 4) × a² × h

For example, a regular pentagonal prism with a = 5 cm and h = 10 cm has a base area of approximately 43.01 cm² and a volume of approximately 430.12 cm³.

Regular vs Custom Base Area Mode

This calculator offers two input modes:

  • Regular Pentagon mode — Enter the side length of the regular pentagonal base and the height. The base area is derived automatically using the formula above. Use this when all five sides of the base pentagon are equal.
  • Custom Base Area mode — Enter the base area directly and the height. This is ideal when the pentagon is irregular, when the area has been measured or computed elsewhere, or when you know only the cross-sectional area of an extruded object. The volume is then simply V = A × h.

Unit Conversions

The calculator supports five common length units: millimetres (mm), centimetres (cm), metres (m), inches (in), and feet (ft). After computing the volume in your chosen unit, the tool automatically converts the result to all other cubic units — mm³, cm³, m³, in³, and ft³ — displayed in a summary table. This saves extra manual conversion steps when you need to report results in multiple unit systems.

Practical Applications

Pentagonal prisms arise in a wide range of real-world contexts:

  • Architecture and construction — calculating concrete volume for five-sided columns, pillars, and decorative structural elements
  • Optics and instruments — pentagonal prisms and pentaprisms are used in single-lens reflex cameras and surveying instruments to bend light paths by 90 degrees
  • Manufacturing — estimating material volume for pentagonal bar stock, extruded profiles, and injection-moulded parts
  • Packaging — computing the capacity of five-sided containers, boxes, and display packaging
  • Engineering and crystallography — modelling pentagonal cross-section members in structural analysis and certain crystal lattice formations

How to Use This Calculator

Select Regular Pentagon if your base is a regular pentagon and enter the side length a. Select Custom Base Area if you already know the area of the base. Enter the height (the perpendicular distance between the two pentagonal bases), choose your unit, and adjust decimal places as needed. Results update instantly. Click Show Step-by-Step to see the full derivation, and use the Copy All button to copy the complete results including the unit conversion table to your clipboard.

Understanding the Pentagon Area Constant

The coefficient √(25 + 10√5) / 4 ≈ 1.7205 comes from the geometry of a regular pentagon. A regular pentagon with side length a can be split into five congruent isosceles triangles radiating from the centre. Each triangle has base a and a specific height (the apothem). Summing their areas yields the formula above. This constant is exact and applies to any regular pentagon regardless of size or unit.

Frequently Asked Questions

Is the Pentagonal Prism Volume Calculator free?

Yes, Pentagonal Prism Volume Calculator is totally free :)

Can I use the Pentagonal Prism Volume Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Pentagonal Prism Volume Calculator?

Yes, any data related to Pentagonal Prism Volume Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is the formula for the volume of a pentagonal prism?

For a regular pentagonal prism, the volume is V = A × h, where A = (√(25 + 10√5) / 4) × a² is the area of the regular pentagonal base (a = side length) and h is the height of the prism. The constant √(25 + 10√5) / 4 ≈ 1.7205. For a prism with an irregular pentagonal base, you supply the base area directly and the volume is simply V = A × h.

How does this pentagonal prism volume calculator work?

Choose Regular mode and enter the side length of the regular pentagonal base along with the prism height, or switch to Custom Base Area mode and enter a known base area and height. Select a unit, and the calculator instantly computes the volume with a step-by-step breakdown. A multi-unit conversion table shows the result in mm³, cm³, m³, in³, and ft³.

What is the difference between Regular mode and Custom Base Area mode?

Regular mode is for prisms whose pentagonal base is a regular pentagon (all five sides equal). The base area is auto-calculated from the side length using the standard formula. Custom Base Area mode is for any pentagonal prism where the base area is already known or the pentagon is irregular — you enter the area directly without needing individual side lengths.

How accurate are the results?

Calculations use IEEE 754 double-precision floating-point arithmetic, which provides roughly 15–17 significant digits of precision. The primary result is displayed to 4 decimal places and unit conversions use 6 significant figures. For very large or very small inputs, switch to a more appropriate unit to avoid rounding artefacts in the display.

What are some real-world examples of pentagonal prisms?

Pentagonal prisms appear in architecture (pentagon-shaped columns and pilasters), packaging (five-sided gift boxes and containers), optics (prism glass elements used in binoculars and cameras), engineering (pentagonal bar stock and extrusions), and crystallography (certain mineral crystal formations). The Pentagon building in Washington D.C. is a famous pentagonal footprint, though not a prism in the strict geometric sense.

Which units does the calculator support?

The calculator supports millimetres (mm), centimetres (cm), metres (m), inches (in), and feet (ft). The selected unit applies to both inputs. Volume results are displayed in the selected unit cubed and the conversion table shows the result simultaneously in all five cubic units.