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Last edited by David Eppstein (talk | contribs) 13 hours ago. (Update)

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Comment: This needs published sources that provide significant depth of coverage of this specific shape, separately from all other related polyhedra, in order to show that it meets our standards for independent notability. Also the AI slop conclusion needs to go. —David Eppstein (talk) 21:55, 16 April 2025 (UTC)

Elongated Decagonal Pyramid

The elongated decagonal pyramid is a type of geometric figure that belongs to the family of pyramids with a base in the shape of a decagon. This solid is characterized by having ten triangular faces that converge at a single apex, as well as an elongated structure which differentiates it from a regular decagonal pyramid. The elongated decagonal pyramid finds applications in various fields, including architecture, mathematics, and computer graphics.

Geometry

Definition

An elongated decagonal pyramid consists of two parallel decagonal bases and ten triangular lateral faces that connect corresponding edges of the two bases. This creates an elongated shape, which can be visualized as a regular decagonal pyramid stretched along its height, with the top base being congruent to the bottom base.

Properties

Vertices: The elongated decagonal pyramid has a total of 22 vertices—10 from the bottom base, 10 from the top base, and 2 additional vertices at the apex.
Edges: It contains 30 edges, comprising 10 edges from the bottom base, 10 edges from the top base, and 10 lateral edges connecting the vertices of the two bases.
Faces: The figure has 12 faces—2 decagonal bases and 10 triangular lateral faces.

Dimensions

The dimensions of an elongated decagonal pyramid can be described using various parameters:

Base Edge Length (a): The length of one side of the regular decagonal base.
Height (h): The perpendicular distance between the two decagonal bases.
Slant Height (l): The distance from the apex to the midpoint of any edge of the base, calculated using the Pythagorean theorem.

Mathematical Formulation

The volume V of an elongated decagonal pyramid can be calculated using the formula:

$V=31*A_{b}*h$

where $A_{b}$ is the area of the decagonal base. The surface area $S$ can be calculated as:

$S=2A_{b}+A_{l}$

where $A_{l}$ is the area of the lateral faces.

Applications

Elongated decagonal pyramids are often used in architecture for designing stepped structures and can also be found in various artistic representations. Additionally, they serve as subjects in mathematical studies involving polyhedra and can be utilized in computer graphics for modeling complex shapes.

Related Geometric Shapes

The elongated decagonal pyramid is related to several other geometric forms, including:

Regular Decagonal Pyramid: A pyramid with congruent triangular faces that taper to a single apex without elongation.
Decagonal Prism: A polyhedron with two parallel decagonal faces connected by rectangular lateral faces.

Conclusion

The elongated decagonal pyramid is a unique geometric structure that exemplifies the diversity of polyhedra. Its properties and applications extend into various fields, making it a subject of interest in both mathematical theory and practical design.