Elongated Pentagonal Pyramid Calculator
Calculator and formulas for an elongated pentagonal pyramid (Johnson solid J9)
Elongated Pentagonal Pyramid Calculator
The Elongated Pentagonal Pyramid
The elongated pentagonal pyramid is a Johnson solid (J9) with 5 triangular faces and 5 square faces.
Johnson Solid J9 Properties
The elongated pyramid: Pentagonal pyramid combined with pentagonal prism
Elongated Pentagonal Pyramid Structure
The elongated pyramid with pentagonal base.
Johnson solid J9.
What is an elongated pentagonal pyramid?
The elongated pentagonal pyramid is a fascinating Johnson solid:
- Definition: Pentagonal pyramid combined with pentagonal prism
- Johnson solid: J9 in the classification
- Triangular faces: 5 congruent equilateral triangles
- Square faces: 5 congruent squares
- Base: 1 regular pentagon
- Edges: 15 edges (all equal length)
Geometric Properties of the Elongated Pentagonal Pyramid
The elongated pentagonal pyramid shows remarkable geometric properties:
Basic Parameters
- Triangular faces: 5 equilateral triangles
- Square faces: 5 congruent squares
- Pentagonal base: 1 regular pentagon
- Vertices: 11 vertices total
Special Properties
- Pentagonal symmetry: 5-fold rotational symmetry
- Golden ratio: Contains φ relationships
- Uniform edges: All edges equal length
- Convex: No concave edges or vertices
Mathematical Relationships
The elongated pentagonal pyramid follows complex pentagonal laws:
Volume Formula
Complex formula involving golden ratio. Contains √5 relationships.
Surface Area Formula
Combination of pentagon, squares, and triangles. Complex √5 and √3 relationships.
Applications of the Elongated Pentagonal Pyramid
Elongated pentagonal pyramids find applications in various fields:
Architecture & Construction
- Pentagonal tower designs
- Unique roof structures
- Architectural spires
- Modern building elements
Science & Technology
- Crystal structures
- Molecular geometry
- Quasicrystal research
- Materials science
Education & Teaching
- Golden ratio studies
- Pentagonal geometry
- Johnson solid education
- Advanced mathematics
Art & Design
- Pentagonal sculptures
- Geometric art pieces
- Golden ratio art
- Contemporary design
Elongated Pentagonal Pyramid Formulas
Volume (V)
Complex volume formula with golden ratio relationships
Surface Area (S)
Pentagon base plus 5 squares and 5 triangles
Height (h)
Height with golden ratio factor
Johnson Solid
9th Johnson solid in the classification
Elongated Pentagonal Pyramid Parameters
5 equilateral △
5 squares □
1 pentagon ⬟
11 vertices
15 edges
All properties follow from the pentagonal symmetry and golden ratio relationships
Calculation Example for an Elongated Pentagonal Pyramid
Given
Find: All properties of the elongated pentagonal pyramid
1. Volume Calculation
For a = 10:
\[V = \frac{5+\sqrt{5}+6\sqrt{25+10\sqrt{5}}}{24} \cdot 10^3\] \[V ≈ 2.022 \cdot 1000\] \[V ≈ 2022\]The volume is approximately 2022 cubic units
2. Surface Area Calculation
For a = 10:
\[S = \frac{20+5\sqrt{3}+\sqrt{25+10\sqrt{5}}}{4} \cdot 10^2\] \[S ≈ 8.8854 \cdot 100\] \[S ≈ 888.54\]The surface area is approximately 888.54 square units
3. Height Calculation
For a = 10:
\[h = \left(1+\frac{5-\sqrt{5}}{10}\right) \cdot 10\] \[h ≈ 1.5257 \cdot 10\] \[h ≈ 15.257\]The height is approximately 15.257 length units
4. The Pentagonal Structure
The elongated pentagonal pyramid with golden ratio symmetry
The Elongated Pentagonal Pyramid: Golden Ratio Geometry
The elongated pentagonal pyramid is one of the most mathematically intriguing Johnson solids, embodying the profound beauty of pentagonal symmetry and golden ratio relationships. By combining a pentagonal pyramid with a pentagonal prism, this structure creates a unique geometry that connects to some of the deepest mathematical constants in nature. The formulas involve complex expressions with √5, revealing the underlying golden ratio φ = (1+√5)/2 that governs pentagonal geometry. As Johnson solid J9, it represents the elegant fusion of regular pentagons, squares, and equilateral triangles in perfect harmony.
The Geometry of the Golden Ratio
The elongated pentagonal pyramid showcases the beauty of pentagonal geometry:
- Pentagonal base: Regular pentagon with internal golden ratio relationships
- Golden ratio connections: All measurements involve φ = (1+√5)/2
- 5-fold symmetry: Pentagonal rotational symmetry
- Johnson solid: J9 in the classical classification
- Mixed faces: Pentagon, squares, and triangles in harmony
- Uniform edges: All edges have the same length
- Natural occurrence: Related to quasicrystal structures
Mathematical Elegance
Golden Ratio Perfection
The complex formulas beautifully encode the golden ratio φ, showing how pentagonal geometry naturally leads to the most aesthetically pleasing proportions in mathematics.
Pentagonal Harmony
The 5-fold symmetry creates a perfect balance between the regularity of squares and triangles with the natural beauty of the pentagon.
Complex Mathematics
The formulas involve nested square roots and golden ratio terms, demonstrating the deep mathematical complexity hidden in seemingly simple geometric forms.
Natural Connections
The pentagonal symmetry connects to patterns found throughout nature, from flowers to crystalline structures and even quasicrystals.
Summary
The elongated pentagonal pyramid stands as a testament to the profound beauty of mathematical relationships in geometry. Its intricate formulas, governed by the golden ratio and √5 expressions, reveal the deep connections between simple geometric construction and fundamental mathematical constants. As Johnson solid J9, it bridges the gap between elementary geometry and advanced mathematical theory, showcasing how pentagonal symmetry leads to some of the most elegant and complex relationships in three-dimensional geometry. From theoretical mathematics to crystallography and beyond, this remarkable polyhedron continues to inspire with its perfect blend of simplicity and sophistication.
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