Elongated Triangular Pyramid Calculator
Calculator and formulas for an elongated triangular pyramid (Johnson solid J7)
Elongated Triangular Pyramid Calculator
The Elongated Triangular Pyramid
The elongated triangular pyramid is a Johnson solid (J7) with 3 triangular faces and 3 square faces.
Johnson Solid J7 Properties
The elongated pyramid: Triangular pyramid combined with triangular prism
Elongated Triangular Pyramid Structure
The elongated pyramid with triangular base.
Johnson solid J7.
What is an elongated triangular pyramid?
The elongated triangular pyramid is a fascinating Johnson solid:
- Definition: Triangular pyramid combined with triangular prism
- Johnson solid: J7 in the classification
- Triangular faces: 3 congruent equilateral triangles
- Square faces: 3 congruent squares
- Base: 1 equilateral triangle
- Edges: 9 edges (all equal length)
Geometric Properties of the Elongated Triangular Pyramid
The elongated triangular pyramid shows remarkable geometric properties:
Basic Parameters
- Triangular faces: 3 equilateral triangles
- Square faces: 3 congruent squares
- Triangle base: 1 equilateral triangle
- Vertices: 7 vertices total
Special Properties
- Triangular symmetry: 3-fold rotational symmetry
- Minimal structure: Simplest elongated pyramid
- Uniform edges: All edges equal length
- Convex: No concave edges or vertices
Mathematical Relationships
The elongated triangular pyramid follows elegant triangular laws:
Volume Formula
Complex formula with √2 and √3. Elegant triangular relationship.
Surface Area Formula
4 triangles plus 3 squares. Simple √3 relationship.
Applications of the Elongated Triangular Pyramid
Elongated triangular pyramids find applications in various fields:
Architecture & Construction
- Triangular tower designs
- Minimal roof structures
- Space-efficient designs
- Structural elements
Science & Technology
- Crystal structures
- Molecular geometry
- Optical applications
- Material science
Education & Teaching
- Triangular geometry studies
- 3D visualization
- Johnson solid education
- Basic polyhedron theory
Art & Design
- Minimalist sculptures
- Triangular art pieces
- Modern design elements
- Geometric patterns
Elongated Triangular Pyramid Formulas
Volume (V)
Volume combining triangular prism and pyramid
Surface Area (S)
4 triangles plus 3 squares
Height (h)
Height with √6 factor for triangular geometry
Johnson Solid
7th Johnson solid in the classification
Elongated Triangular Pyramid Parameters
3 equilateral △
3 squares □
1 triangle ▲
7 vertices
9 edges
All properties follow from the triangular symmetry and √2, √3, √6 relationships
Calculation Example for an Elongated Triangular Pyramid
Given
Find: All properties of the elongated triangular pyramid
1. Volume Calculation
For a = 10:
\[V = \frac{\sqrt{2}+3\sqrt{3}}{12} \cdot 10^3\] \[V ≈ 0.5509 \cdot 1000\] \[V ≈ 550.9\]The volume is approximately 550.9 cubic units
2. Surface Area Calculation
For a = 10:
\[S = (3+\sqrt{3}) \cdot 10^2\] \[S ≈ 4.732 \cdot 100\] \[S ≈ 473.2\]The surface area is approximately 473.2 square units
3. Height Calculation
For a = 10:
\[h = \left(1+\frac{\sqrt{6}}{3}\right) \cdot 10\] \[h ≈ 1.8165 \cdot 10\] \[h ≈ 18.165\]The height is approximately 18.165 length units
4. The Triangular Structure
The elongated triangular pyramid with perfect triangular symmetry
The Elongated Triangular Pyramid: Triangular Simplicity
The elongated triangular pyramid represents the most elegant and minimal example of elongated pyramid geometry. By combining a triangular pyramid with a triangular prism, this Johnson solid creates the simplest possible three-dimensional extension of triangular symmetry. The mathematical beauty lies in the harmonious interplay of √2, √3, and √6 relationships that govern all proportions, reflecting the fundamental geometry of the equilateral triangle. As Johnson solid J7, it demonstrates how triangular base structures can be elegantly extended while maintaining perfect geometric harmony and structural integrity.
The Geometry of Triangular Minimalism
The elongated triangular pyramid showcases the beauty of triangular-based geometry:
- Triangular base: Perfect equilateral triangle foundation
- 3-fold symmetry: 120° rotational symmetry
- Minimal structure: Simplest possible elongated pyramid
- Johnson solid: J7 in the classical classification
- Mixed faces: Triangles and squares in perfect harmony
- Uniform edges: All edges have the same length
- Structural efficiency: Maximum strength with minimum material
Mathematical Elegance
Square Root Harmony
The formulas beautifully combine √2, √3, and √6 factors, showing how triangular geometry naturally leads to these fundamental mathematical relationships.
Minimal Complexity
Despite being the simplest elongated pyramid, it contains rich mathematical relationships that connect to deeper geometric principles.
Structural Elegance
The triangular base provides excellent structural stability while using the minimum number of faces and vertices possible for an elongated pyramid.
Geometric Purity
The perfect balance between triangular and square faces creates a visually harmonious form that exemplifies geometric purity and mathematical beauty.
Summary
The elongated triangular pyramid stands as a testament to the power of geometric minimalism and mathematical elegance. Its simple yet sophisticated structure, governed by elegant square root relationships, makes it both theoretically fascinating and practically valuable. As Johnson solid J7, it represents the perfect entry point into understanding elongated pyramid geometry while demonstrating that simplicity and mathematical beauty are not mutually exclusive. From basic geometric education to advanced architectural applications, this remarkable polyhedron continues to inspire with its perfect balance of minimalism and sophistication, proving that sometimes the most elegant solutions are also the simplest.
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