Oblique Prism Calculator

Calculator and formulas for an oblique prism

The Slanted Prism - Where Geometry Meets Perspective!

Oblique Prism Calculator

The Oblique Prism

An oblique prism is a slanted prismatic solid where the top surface is shifted horizontally, creating parallelogram side faces.

Enter parameters
Vertices of base polygon
Side length of base polygon
What parameter do you know?
Value of selected parameter
Oblique angle in degrees
Oblique Prism Calculation Results
Height (h):
Length (b):
Base area (A):
Surface:
Volume:
Oblique Prism Properties

The slanted prism: Top surface shifted horizontally creating parallelogram sides

Slanted geometry Parallelogram faces Non-perpendicular Same base area

Oblique Prism Structure

Oblique Prism

The prism with shifted parallel bases.
Creates parallelogram side faces.

What is an Oblique Prism?

An oblique prism represents a fascinating variation of the traditional prism:

  • Definition: Prism with shifted parallel bases
  • Top surface: Horizontally shifted relative to base
  • Side faces: Parallelograms instead of rectangles
  • Volume: Same as right prism (Cavalieri's principle)
  • Angle: Characterized by oblique angle α
  • Applications: Architecture and engineering

Geometric Properties of the Oblique Prism

The oblique prism displays unique slanted characteristics:

Oblique Parameters
  • Bases: Parallel but shifted polygons
  • Side faces: Parallelograms
  • Height: Perpendicular distance between bases
  • Length b: Actual edge length of parallelogram
Special Properties
  • Cavalieri's principle: Same volume as right prism
  • Oblique angle: Determines degree of slant
  • Cross-section: Uniform throughout height
  • Trigonometry: Height = b × sin(α)

Mathematical Relationships

The oblique prism follows elegant trigonometric principles:

Trigonometric Relations
h = b × sin(α)

Height relates to edge length via sine. Simple trigonometric relationship.

Cavalieri's Principle
V = A × h

Volume equals base area times height. Same as right prism formula.

Applications of the Oblique Prism

Architecture & Construction
  • Slanted building facades
  • Angled roof structures
  • Modern architectural design
  • Leaning structural elements
Engineering & Manufacturing
  • Angled mechanical parts
  • Sloped structural beams
  • Inclined conveyor systems
  • Perspective engineering
Education & Mathematics
  • Cavalieri's principle demonstration
  • Trigonometry applications
  • 3D geometry studies
  • Perspective drawing
Art & Design
  • Artistic sculptures
  • Perspective illustrations
  • Modern furniture design
  • Geometric art installations

Formula for the oblique prism

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

Area of regular n-sided polygon base

Height (h)
\[h = b \cdot \sin(\alpha)\]

Perpendicular height from oblique edge

Length (b)
\[b = \frac{h}{\sin(\alpha)}\]

Oblique edge length from height

Surface area (S)
\[S = 2 \cdot A + n \cdot a \cdot h\]

Two bases plus n parallelogram faces

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

Volume by Cavalieri's principle: base area times perpendicular height

Example calculation for an Oblique Prism

Given
Vertices n = 5 Base length a = 6 Height h = 10 Angle α = 75°

Calculate: All properties of the pentagonal oblique prism

1. Base area calculation
\[A = \frac{5 \times 6^2}{4 \times \tan(36°)}\] \[A = \frac{180}{4 \times 0.727} = 61.93\]

The base area is 61.93 square units

2. Edge length calculation
\[b = \frac{10}{\sin(75°)}\] \[b = \frac{10}{0.966} = 10.35\]

The oblique edge length is 10.35 units

3. Volume calculation
\[V = 61.93 \times 10 = 619.3\]

The volume is 619.3 cubic units

4. Surface area calculation
\[S = 2 \times 61.93 + 5 \times 6 \times 10\] \[S = 123.86 + 300 = 423.86\]

The surface area is 423.86 square units

Summary
Base: 61.93 Height: 10.0
Length: 10.35 Angle: 75°
Volume: 619.3 Surface: 423.86
⚡ Slanted

Complete analysis of the pentagonal oblique prism

The Oblique Prism: Where Geometry Meets Perspective

The oblique prism represents a fascinating exploration into non-perpendicular geometry, where traditional rigid vertical structures give way to slanted, dynamic forms. This geometric variation demonstrates how shifting parallel bases creates entirely new spatial relationships while maintaining fundamental volume properties. Through the elegant application of Cavalieri's principle and trigonometric relationships, the oblique prism bridges the gap between theoretical geometry and practical applications in architecture, engineering, and design.

The Beauty of Slanted Geometry

The oblique prism showcases the elegance of non-perpendicular design:

  • Cavalieri's principle: Volume remains constant regardless of slant angle
  • Trigonometric harmony: Height and edge length related by sine function
  • Parallelogram faces: Creates dynamic visual perspective
  • Architectural freedom: Enables slanted and angled designs
  • Engineering flexibility: Accommodates non-vertical structural needs
  • Mathematical elegance: Simple formulas with profound implications
  • Design innovation: Opens new possibilities in spatial composition

Mathematical Principles and Real-World Impact

Cavalieri's Revolutionary Insight

Bonaventura Cavalieri's principle revolutionized our understanding of volume, showing that slanting doesn't change volume - a principle fundamental to the oblique prism's properties.

Trigonometric Relationships

The simple relationship h = b × sin(α) demonstrates how trigonometry governs the transition from slanted edge to vertical height, essential for practical calculations.

Architectural Innovation

Modern architecture extensively uses oblique forms to create dynamic spaces and striking visual effects, from leaning towers to angled facades.

Engineering Applications

In engineering, oblique prisms model everything from angled structural supports to inclined conveyor systems, where non-vertical orientations are essential.

Conclusion

The oblique prism stands as a testament to geometry's adaptability and practical relevance. By breaking free from perpendicular constraints, it opens doors to innovative design while maintaining mathematical rigor through Cavalieri's principle and trigonometric relationships. From the slanted walls of modern buildings to the angled components of complex machinery, oblique prisms demonstrate how mathematical concepts directly enable real-world innovations. For students and professionals alike, the oblique prism serves as an excellent example of how theoretical mathematics translates into practical applications, showing that even simple geometric variations can have profound implications for design, engineering, and our understanding of three-dimensional space. The oblique prism truly embodies the perfect fusion of mathematical principle and creative possibility.