Parallelepiped Calculator

Calculator and formulas to calculate a parallelepiped

The Skewed Box - Six Parallelograms in Perfect Harmony!

Parallelepiped Calculator

The Parallelepiped

A parallelepiped is a three-dimensional figure formed by six parallelograms, where opposite faces are parallel and congruent.

Enter Parameters
First side dimension
Second side dimension
Third side dimension
Angle between sides b and c
Angle between sides a and c
Angle between sides a and b
Parallelepiped Calculation Results
Surface S:
Volume V:
Height h:
Parallelepiped Properties

The skewed box: Six parallelograms with opposite faces parallel

6 Parallelogram faces 12 Edges 8 Vertices 3 Pairs of parallel faces

Parallelepiped Structure

Parallelepiped

The skewed box with six parallelogram faces.
Opposite faces are parallel and congruent.

What is a Parallelepiped?

A parallelepiped is a fascinating three-dimensional geometric solid:

  • Definition: Solid bounded by six parallelograms
  • Parallel faces: Opposite faces are parallel and congruent
  • Structure: Three pairs of parallel faces
  • Vertices: 8 vertices total
  • Edges: 12 edges (4 groups of 3 parallel edges)
  • Special cases: Cuboids and rhombohedra

Geometric Properties of the Parallelepiped

The parallelepiped exhibits remarkable geometric properties:

Basic Parameters
  • Faces: 6 parallelograms
  • Edge structure: 4 groups of 3 parallel edges
  • Euler formula: V - E + F = 8 - 12 + 6 = 2
  • Angles: Three independent angles α, β, γ
Special Properties
  • Affine transformation: Any parallelepiped can be transformed to any other
  • Volume invariant: Volume preserved under shearing
  • Diagonal symmetry: Space diagonals bisect each other
  • Parallelism: Opposite faces are parallel

Mathematical Relationships

The parallelepiped follows elegant trigonometric laws:

Volume Formula
Complex trigonometric expression

Involves all three sides and angles. Based on scalar triple product.

Surface Formula
Sum of parallelogram areas

Sum of six parallelogram faces. Uses sine functions for areas.

Applications of Parallelepipeds

Parallelepipeds find applications in various fields:

Architecture & Construction
  • Skewed building blocks
  • Modern architectural designs
  • Structural elements
  • Non-orthogonal spaces
Science & Technology
  • Crystal lattice structures
  • Linear algebra applications
  • Vector space geometry
  • Materials science
Education & Teaching
  • 3D geometry education
  • Vector mathematics
  • Trigonometry applications
  • Linear transformation studies
Art & Design
  • Sculptural works
  • Geometric installations
  • 3D printing projects
  • Architectural models

Parallelepiped Formulas

Volume (V)
\[V=a\cdot b\cdot c\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}\]

Complex trigonometric volume formula

Surface (S)
\[S=2\cdot a\cdot b\cdot \sin(\gamma )+2\cdot a\cdot c\cdot \sin(\beta )+2\cdot b\cdot c\cdot \sin(\alpha )\]

Sum of three pairs of parallelogram areas

Height (h)
\[h={\frac {a}{\sin(\alpha )}}\cdot {\sqrt {1+2\cdot \cos(\alpha )\cdot \cos(\beta )\cdot \cos(\gamma )-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )}}\]

Height perpendicular to base parallelogram

Note
Calculator in DEGREE mode!

Angle functions require degree mode

Parallelepiped Parameters
Faces
6 parallelograms
Edges
12 edges total
Vertices
8 vertices
Angles
α, β, γ independent

All properties depend on the three side lengths and three angles

Calculation Example for a Parallelepiped

Given
Side a = 7 Side b = 8 Side c = 9 Angle α = 30° Angle β = 40° Angle γ = 50°

Find: All properties of the parallelepiped

1. Volume Calculation

First calculate the radical:

\[\sqrt{1 + 2\cos(30°)\cos(40°)\cos(50°) - \cos^2(30°) - \cos^2(40°) - \cos^2(50°)}\] \[≈ \sqrt{1 + 0.678 - 0.750 - 0.587 - 0.413}\] \[≈ \sqrt{-0.072} \text{ (imaginary!)}\]

Note: These angles would create an impossible parallelepiped!

2. Valid Example

With angles 60°, 70°, 80°:

\[V = 7 \cdot 8 \cdot 9 \cdot \sqrt{1 + 2(0.5)(0.342)(0.174) - 0.25 - 0.117 - 0.030}\] \[V ≈ 504 \cdot \sqrt{0.663}\] \[V ≈ 410.2\]

Volume is approximately 410.2 cubic units

3. The Valid Parallelepiped (a=7, b=8, c=9, α=60°, β=70°, γ=80°)
Volume V ≈ 410.2 Surface S ≈ 398.4 Height h ≈ 7.3
6 Parallelogram faces 12 Equal opposite edges 8 Vertices
Non-orthogonal Skewed geometry 📐 Valid angles

A valid parallelepiped with proper angle constraints

The Parallelepiped: The Skewed Box

The parallelepiped is a fundamental geometric solid that generalizes the concept of a rectangular box to non-orthogonal space. With its six parallelogram faces arranged in three pairs of parallel surfaces, it represents one of the most important shapes in linear algebra and vector geometry. The mathematical beauty lies in the elegant relationship between its three side vectors and the resulting volume, which is computed using the scalar triple product - a cornerstone of vector calculus.

The Geometry of Parallel Faces

The parallelepiped demonstrates the perfection of parallel geometry:

  • Parallel faces: Opposite faces are parallel and congruent parallelograms
  • Edge structure: 12 edges arranged in 4 groups of 3 parallel edges each
  • Vector representation: Defined by three linearly independent vectors
  • Affine invariance: Shape preserved under affine transformations
  • Volume formula: Absolute value of the scalar triple product
  • Special cases: Includes rectangular boxes, rhombohedra, and cubes
  • Constraints: Not all angle combinations are geometrically possible

Mathematical Elegance

Vector Space Foundation

The parallelepiped is the geometric embodiment of a vector space basis, where three linearly independent vectors span a three-dimensional space.

Trigonometric Complexity

The volume formula involves a sophisticated trigonometric expression that ensures the result is always real and positive for valid geometries.

Linear Algebra Applications

Parallelepipeds are essential in understanding determinants, matrix theory, and coordinate transformations in higher mathematics.

Crystal Lattices

The fundamental building blocks of crystal structures are parallelepipeds, making this shape crucial in materials science and crystallography.

Summary

The parallelepiped stands as a bridge between elementary geometry and advanced mathematics. Its deceptively simple structure - six parallelogram faces - conceals deep mathematical relationships that connect vector algebra, trigonometry, and linear transformations. From its role as the fundamental domain in crystal lattices to its application in computer graphics and architectural design, the parallelepiped demonstrates how geometric forms can embody both mathematical elegance and practical utility. As students progress from basic box shapes to understanding the full generality of skewed parallelepipeds, they gain insight into the beautiful complexity that underlies our three-dimensional world.