Pentagonal Bipyramid Calculator
Calculator and formulas for calculating a regular pentagonal bipyramid (or dipyramid)
Pentagonal Bipyramid Calculator
The Pentagonal Bipyramid
A pentagonal bipyramid consists of 10 congruent isosceles triangles around a regular pentagon as base.
Pentagonal Bipyramid Properties
The golden ratio bipyramid: Two pyramids connected at pentagon base
Pentagonal Bipyramid Structure
The pentagonal bipyramid with perfect symmetry.
10 isosceles triangles around a pentagon.
What is a pentagonal bipyramid?
A pentagonal bipyramid is a fascinating geometric solid:
- Definition: Two pyramids connected at pentagon base
- Base: Regular pentagon as central foundation
- Faces: 10 congruent isosceles triangles
- Vertices: 7 vertices (5 equatorial, 2 polar)
- Edges: 15 edges (5+5+5)
- Symmetry: Perfect D5h-symmetry
Geometric Properties of the Pentagonal Bipyramid
The pentagonal bipyramid shows remarkable geometric properties:
Basic Parameters
- Faces: 10 isosceles triangles
- Vertices: 7 vertices (5 equatorial, 2 polar)
- Edges: 15 edges (all equal length)
- Euler characteristic: V - E + F = 7 - 15 + 10 = 2
Special Properties
- Deltahedron: All faces are triangles
- Bipyramid: Two mirrored pyramids
- Pentagon base: Golden ratio connection
- Convex: No inward-pointing edges
Mathematical Relationships
The pentagonal bipyramid follows elegant mathematical laws:
Volume Formula
Uses the tangent function for π/5. Elegant and precise.
Surface Formula
10 congruent isosceles triangles. Five-fold symmetry.
Applications of the Pentagonal Bipyramid
Pentagonal bipyramids find applications in various fields:
Architecture & Construction
- Church spires and towers
- Decorative roof elements
- Structural bracing
- Sculptural building components
Science & Technology
- Crystallographic structures
- Molecular geometry
- Optical prisms
- Mechanical components
Education & Teaching
- Geometry lessons
- 3D geometry studies
- Symmetry demonstrations
- Polyhedron classification
Art & Design
- Geometric sculptures
- Modern artworks
- Decorative objects
- Jewelry design
Formulas for the Pentagonal Bipyramid
Slant Height (s)
Slant height with cot(π/5) ≈ 1.376
Edge Length (e)
Length of the bipyramid edges
Total Height (i)
Double the height of single pyramid
Slant Area (As)
Area of one isosceles triangle
Surface Area (S)
10 isosceles triangular faces
Perimeter (P)
Perimeter of the pentagon
Volume (V)
With π/5 factor through tan(π/5) ≈ 0.727
Calculation Example for a Pentagonal Bipyramid
Given
Find: All properties of the pentagonal bipyramid
1. Slant Height Calculation
For pentagon (cot(π/5) ≈ 1.376):
\[s = \sqrt{10^2 + \frac{8^2 \cdot 1.376^2}{4}}\] \[s = \sqrt{100 + \frac{64 \cdot 1.894}{4}}\] \[s = \sqrt{100 + 30.30} ≈ 11.43\]The slant height is approximately 11.43 units
2. Volume Calculation
With tan(π/5) ≈ 0.727:
\[V = \frac{5 \cdot 8^2 \cdot 10}{6 \cdot 0.727}\] \[V = \frac{5 \cdot 64 \cdot 10}{4.362}\] \[V ≈ 733.1\]The volume is approximately 733.1 cubic units
3. Surface Area Calculation
10 triangles:
\[S = 5 \cdot a \cdot s\] \[S = 5 \cdot 8 \cdot 11.43\] \[S ≈ 457.2\]The surface area is approximately 457.2 square units
4. The Perfect Pentagonal Bipyramid
The pentagonal bipyramid with perfect symmetry
The Pentagonal Bipyramid: The Golden Ratio in Perfect Symmetry
The pentagonal bipyramid is a fascinating geometric solid that embodies the elegance of five-fold symmetry. By connecting two pyramids at a regular pentagon base, it creates a unique structure with 10 congruent isosceles triangles, making this solid one of the most beautiful examples of geometric perfection. The mathematical beauty lies in the simple yet elegant relationships with the trigonometric functions of the π/5 angle, which connects all geometric properties and relates to the golden ratio through the pentagon's inherent proportions.
The Geometry of Five-Fold Perfection
The pentagonal bipyramid shows the perfection of pentagonal symmetry:
- Deltahedron: All 10 faces are congruent isosceles triangles
- D5h-symmetry: Five-fold rotational symmetry with mirror plane
- Golden ratio connection: Pentagon base relates to φ = (1+√5)/2
- Uniformity: All 15 edges have the same length
- Bipyramid structure: Two mirrored pyramids in perfect harmony
- Convexity: All vertices point outward
- Versatility: Ideal for constructions and applications
Mathematical Elegance
π/5 Trigonometry
The formulas of the pentagonal bipyramid are masterpieces of simplicity, with π/5 trigonometric functions as elegant factors that describe the geometric relationships of the pentagon.
Golden Ratio Connection
As a combination of pentagon-based pyramids, it shows the relationship to the golden ratio and its harmonic proportions found throughout nature.
Structural Perfection
The perfect symmetry and stability make the bipyramid a preferred form in nature and technology applications.
Aesthetic Completion
The harmonic union of two pyramids over a pentagon creates a unique visual balance between simplicity and complexity.
Summary
The pentagonal bipyramid embodies the perfect balance between mathematical simplicity and geometric beauty. Its structure of ten isosceles triangles, described by elegant π/5 relationships, makes it a fascinating study object for mathematicians, architects, and designers. The natural relationship to the golden ratio through its pentagon base shows the universal significance of five-fold symmetry in nature and mathematics. From pure mathematics to practical applications, the pentagonal bipyramid remains a fascinating example of the power of geometric transformation and the beauty of perfect symmetry in three dimensions.