Regular prism volume calculator

What is a regular prism?

A regular prism is a three-dimensional geometric figure with two congruent polygonal bases connected by rectangular faces. The term “regular” indicates that the polygonal base is a regular polygon, meaning all its sides and interior angles are equal. Common examples include triangular prisms (base: triangle), pentagonal prisms (base: pentagon), and hexagonal prisms (base: hexagon). The volume of a prism depends on the area of its base and its height (the perpendicular distance between the two bases).

Formula for calculating the volume of a regular prism

The volume $V$ of a regular prism is calculated using the formula:

Where:

• $A$ = Area of the base polygon
• $l$ = Height (or length) of the prism (distance between the bases)

For a regular polygon with $n$ sides, each of length $s$, the area $A$ is given by:

Here, $a$ is the apothem (the distance from the center of the polygon to the midpoint of one of its sides). The apothem can be calculated if the side length $s$ is known:

Substituting this into the area formula:

Thus, the final volume formula becomes:

Examples of volume calculations

Example 1: Pentagonal prism

Problem: A regular pentagonal prism has a side length $s = 6 \, \text{cm}$ and height $l = 15 \, \text{cm}$. Calculate its volume.
Solution:

1. Calculate the apothem $a$:

2. Calculate the base area $A$:

3. Calculate the volume $V$:

Example 2: Hexagonal prism

Problem: A regular hexagonal prism has a side length $s = 10 \, \text{cm}$, apothem $a = 8.66 \, \text{cm}$, and height $l = 20 \, \text{cm}$. Find its volume.
Solution:

1. Calculate the base area $A$:

2. Calculate the volume $V$:

Example 3: Triangular prism

Problem: A regular triangular prism has a side length $s = 4 \, \text{m}$ and height $l = 10 \, \text{m}$. Determine its volume.
Solution:

1. Calculate the apothem $a$:

2. Calculate the base area $A$:

3. Calculate the volume $V$:

Historical context

The study of prisms dates back to ancient Greece, where mathematicians like Euclid explored their properties in Elements. Regular prisms were also used in architecture; for example, hexagonal columns were employed in Roman and Gothic structures for their structural efficiency. The term “prism” itself originates from the Greek word prisma, meaning “something sawed.”

Frequently Asked Questions

How to calculate the volume of a prism if the apothem is unknown?

Use the formula involving the side length $s$:

For a hexagonal prism ($n = 6$) with $s = 5 \, \text{cm}$ and $l = 12 \, \text{cm}$:

How does the number of sides $n$ affect the volume?

As $n$ increases, the base polygon approximates a circle, and the prism resembles a cylinder. For example, a 100-sided prism’s volume would be close to $\pi r^2 l$, where $r$ is the radius of the circumscribed circle. For calculating the volume of a cylinder, use our cylinder volume calculator.

What is the volume of an octagonal prism with side length 5 cm and height 12 cm?

Using $n = 8$:

How to convert volume from cubic meters to liters?

1 cubic meter ($\text{m}^3$) = 1000 liters. For example, $2.5 \, \text{m}^3 = 2500 \, \text{L}$. For converting different volume units, use our volume converter.