Regular Prism Calculator

Online calculator and formulas for calculating a regular prism

The Perfect Prismatic Form - Parallel Faces in Harmony!

Regular Prism Calculator

The Regular Prism

A Regular Prism is a prismatic body with parallel regular polygonal faces.

Enter Known Parameters
Number of vertices of the base
Side length of the base polygon
What to calculate?
Value of the selected parameter
Regular Prism Calculation Results
Height:
Surface Area:
Base Area:
Regular Prism Properties

The perfect prismatic form: Two parallel regular polygonal faces

n Rectangular side faces 2 Regular n-gons 3n Edges 2n Vertices

Regular Prism Structure

Regular Prism

The perfect prismatic form with parallel faces.
Uniform height throughout.

What is a Regular Prism?

A Regular Prism is a fundamental prismatic solid:

  • Definition: Prism with regular polygonal bases
  • Parallel faces: Two identical parallel polygons
  • Side faces: n congruent rectangles
  • Vertices: 2n vertices total
  • Edges: 3n edges (n+n+n)
  • Uniformity: Constant cross-section

Geometric Properties of Regular Prisms

The Regular Prism demonstrates remarkable geometric properties:

Basic Parameters
  • Base faces: 2 regular n-gons
  • Side faces: n rectangles
  • Euler characteristic: V - E + F = 2n - 3n + (n+2) = 2
  • Cross-section: Constant throughout height
Special Properties
  • Translation symmetry: Along the height axis
  • Rotation symmetry: n-fold around central axis
  • Reflection symmetry: Multiple planes
  • Right angles: Between bases and sides

Mathematical Relationships

The Regular Prism follows elegant trigonometric laws:

Volume Formula
Base area × Height

Simple multiplication of base area and height. Clean and elegant.

Surface Formula
2 × Base + Lateral surface

Sum of two bases and rectangular sides. Elegant geometric form.

Applications of Regular Prisms

Regular Prisms find applications in various fields:

Architecture & Construction
  • Building columns and pillars
  • Structural beams
  • Foundation elements
  • Architectural details
Science & Engineering
  • Crystal structures
  • Optical prisms
  • Mechanical components
  • Packaging design
Education & Teaching
  • Geometry instruction
  • Volume calculations
  • 3D geometry studies
  • Mathematical demonstrations
Art & Design
  • Sculptural works
  • Decorative elements
  • Furniture design
  • Geometric patterns

Formulas for Regular Prism

Surface Area (S)
\[S = \frac{2 \cdot n \cdot a^2}{ 4 \cdot \tan\left(\frac{\pi}{n}\right)} + n \cdot a \cdot h\]

Surface area with tangent formula

Volume (V)
\[V= \frac{n \cdot h \cdot a^2}{ 4 \cdot \tan\left(\frac{\pi}{n}\right)}\]

Volume = Base area × Height

Base Area (A)
\[A = \frac{n \cdot a^2}{ 4 \cdot \tan\left(\frac{\pi}{n}\right)}\]

Area of regular n-gon base

Height (h)
\[h= \frac{4 \cdot V \cdot \tan\left(\frac{\pi}{n}\right)}{n \cdot a^2}\]

Height from volume and base

Regular Prism Parameters
Base faces
2 regular n-gons
Side faces
n rectangles
Vertices
2n vertices
Edges
3n edges

The (n) stands for the number of vertices of the base polygon

Calculation Example for a Regular Prism

Given
Side length a = 5 Vertices n = 5 Volume V = 140

Find: Height of the pentagonal prism

1. Base Area Calculation

For n=5, a=5:

\[A = \frac{5 \cdot 25}{ 4 \cdot \tan(36°)}\] \[A ≈ \frac{125}{4 \cdot 0.727}\] \[A ≈ 43.01\]

The base area is approximately 43.01 square units

2. Height Calculation

Using V = A × h:

\[h = \frac{V}{A} = \frac{140}{43.01}\] \[h ≈ 3.26\]

The height is approximately 3.26 length units

3. The Pentagonal Prism
Side length a = 5.0 Base: Pentagon Base area A ≈ 43.01
Height h ≈ 3.26 Volume V = 140 5 Side faces
15 Edges 10 Vertices ⚡ Perfect Form

The pentagonal prism with perfect parallel symmetry

The Regular Prism: The Perfect Prismatic Form

The Regular Prism is one of the most fundamental and elegant geometric solids, representing the perfect balance between simplicity and structural integrity. With its two parallel regular polygonal faces connected by rectangular sides, the regular prism embodies the essence of prismatic geometry. The mathematical beauty lies in the simple yet powerful relationships between its dimensions, making it both theoretically fascinating and practically invaluable in countless applications.

The Geometry of Perfection

The Regular Prism demonstrates the elegance of parallel geometry:

  • Parallel bases: Two identical regular polygons
  • Uniform cross-section: Constant throughout the height
  • Right angles: Perfect perpendicular relationship
  • Translation symmetry: Invariant under translation
  • Rotation symmetry: n-fold rotational symmetry
  • Reflection symmetry: Multiple reflection planes
  • Structural stability: Inherently strong form

Mathematical Elegance

Simple Formulas

The formulas for regular prisms are elegant in their simplicity, using basic trigonometric functions to relate all geometric properties.

Scalable Design

From triangular to highly polygonal bases, regular prisms scale beautifully while maintaining their fundamental properties.

Practical Applications

The simple geometry makes regular prisms ideal for construction, manufacturing, and countless engineering applications.

Universal Form

Regular prisms appear throughout nature and human design, from crystal structures to architectural elements.

Summary

The Regular Prism stands as a testament to the power of geometric simplicity. Its perfect parallel faces, elegant mathematical relationships, and inherent structural strength make it one of the most important shapes in mathematics, engineering, and design. From ancient architecture to modern manufacturing, the regular prism continues to serve as a foundation of geometric thinking and practical application. Its beauty lies not in complexity, but in the perfect harmony of simple geometric principles working together to create an endlessly useful and aesthetically pleasing form.