Antiprism Calculator

Calculator and formulas for calculating an antiprism

The Twisted Prism - Rotated Base Faces in Perfect Harmony!

Antiprism Calculator

The Antiprism

An Antiprism is a prismatic body with twisted base faces (by 180°/n).

Enter Known Parameters
Which parameter is known?
Value of the selected parameter
Number of vertices of the base (n)
Antiprism Calculation Results
Side length a:
Volume V:
Surface S:
Height h:
Antiprism Properties

The Twisted Prism: Base faces rotated by 180°/n

2n Triangular side faces 2 n-gonal base faces 4n Edges 2n Vertices

Antiprism Structure

Antiprism

The twisted prism with rotated base faces.
Rotation by 180°/n.


What is an Antiprism?

An Antiprism is a fascinating prismatic body:

  • Definition: Prism with twisted base faces
  • Rotation: Top base rotated by 180°/n
  • Side faces: 2n congruent triangles
  • Vertices: 2n vertices total
  • Edges: 4n edges (n+n+2n)
  • Regular: All edges equal length

Geometric Properties of the Antiprism

The Antiprism exhibits remarkable geometric properties:

Basic Parameters
  • Base faces: 2 regular n-gons
  • Side faces: 2n congruent triangles
  • Euler Characteristic: V - E + F = 2n - 4n + (2n+2) = 2
  • Rotation angle: 180°/n
Special Properties
  • Chiral object: Exists in left and right forms
  • Uniform edges: All edges equal length
  • Rotational symmetry: n-fold rotation axis
  • Semiregular: Two types of faces

Mathematical Relationships

The Antiprism follows complex trigonometric laws:

Volume Formula
Complex trigonometric formula

Depends on cos, sin and n. Complex but elegant.

Surface Formula
With cotangent and √3

Sum of base faces and triangles. Elegant trigonometric form.

Applications of the Antiprism

Antiprisms find applications in various fields:

Architecture & Construction
  • Twisted tower structures
  • Modern building designs
  • Structural bracing systems
  • Architectural elements
Science & Technology
  • Crystallographic structures
  • Molecular geometry
  • Optical components
  • Mechanical constructions
Education & Teaching
  • Geometry instruction
  • Trigonometry applications
  • 3D geometry studies
  • Mathematical demonstrations
Art & Design
  • Sculptural works
  • Art installations
  • Decorative objects
  • Geometric patterns

Formulas for the Antiprism

Surface Area (S)
\[S = \frac{n}{2} \cdot \left(\cot\left(\frac{\pi}{n}\right) + \sqrt{3}\right) \cdot a^2\]

Surface area with cotangent and √3

Volume (V)
\[V = \frac{n \cdot \sqrt{4 \cdot \cos^2\left(\frac{\pi}{2n}\right) - 1} \cdot \sin\left(\frac{3\pi}{2n}\right)}{12 \cdot \sin^2\left(\frac{\pi}{n}\right)} \cdot a^3\]

Complex trigonometric volume formula

Height (h)
\[h = \sqrt{1 - \frac{1}{4 \cdot \cos^2\left(\frac{\pi}{2n}\right)}} \cdot a\]

Height dependent on vertex count n

Important Note
Set calculator to RAD mode!

Angle functions require radian mode

Antiprism Parameters
Base faces
2 regular n-gons
Side faces
2n triangles
Rotation
180°/n
Edges
All equal length

All properties depend on the vertex count n of the base

Calculation Example for an Antiprism

Given
Side length a = 10 Vertices n = 5

Find: All properties of the pentagonal antiprism

1. Surface Area Calculation

For n=5:

\[S = \frac{5}{2} \cdot (\cot(36°) + \sqrt{3}) \cdot 100\] \[S ≈ 2.5 \cdot (1.376 + 1.732) \cdot 100\] \[S ≈ 777\]

The surface area is approximately 777 square units

2. Height Calculation

For n=5:

\[h = \sqrt{1 - \frac{1}{4 \cdot \cos^2(18°)}} \cdot 10\] \[h ≈ \sqrt{1 - 0.277} \cdot 10\] \[h ≈ 8.5\]

The height is approximately 8.5 length units

3. The Pentagonal Antiprism
Side length a = 10.0 Base: Pentagon Surface S ≈ 777
Height h ≈ 8.5 10 Triangle faces Rotation: 36°
20 Edges 10 Vertices 🔄 Twisted

The pentagonal antiprism with perfect rotational symmetry

The Antiprism: The Twisted Prism

The Antiprism is a fascinating geometric body that embodies the elegance of rotation. Through the characteristic rotation of the upper base by 180°/n, it creates a unique structure with 2n triangular side faces, making the antiprism one of the most interesting semiregular polyhedra. The mathematical beauty lies in the complex trigonometric relationships that connect all geometric properties, creating a perfect harmony between symmetry, rotation, and mathematical elegance.

The Geometry of Rotation

The antiprism demonstrates the perfection of geometric rotation:

  • Rotation angle: Precisely 180°/n for n-gonal base
  • Chirality: Exists in left and right forms
  • Uniformity: All edges have the same length
  • Semiregularity: Two different face types
  • Symmetry: n-fold rotation axis
  • Trigonometry: Complex sine and cosine relationships
  • Versatility: Works for any vertex count n ≥ 3

Mathematical Elegance

Trigonometric Perfection

The antiprism formulas are masterpieces of trigonometry, showing how sine, cosine, and cotangent work together to describe the twisted geometry.

Parametric Diversity

By varying the vertex count n, infinitely many different antiprisms arise, from triangular to highly polygonal forms with different aesthetic qualities.

Structural Innovation

The rotation creates optimal stress distribution, making antiprisms preferred structures in architecture and engineering.

Aesthetic Perfection

The harmonious combination of regular base faces and uniformly twisted triangles creates unique visual dynamics.

Summary

The Antiprism embodies the perfect balance between mathematical complexity and geometric beauty. Its twisted structure, described by elegant trigonometric formulas, makes it a fascinating study object for mathematicians, architects, and designers. From simple triangular forms to complex polygonal variants, the antiprism shows how rotation of basic forms can lead to entirely new geometric worlds. As a bridge between theoretical mathematics and practical application, the antiprism remains a fascinating example of the power of geometric transformation.