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Calculate volume, surface area, and other properties of a regular pentagonal prism. Enter the side length and height.
A pentagonal prism has two parallel regular pentagon bases connected by five rectangular faces. The regular pentagon has five equal sides and five equal interior angles of 108° each.
The regular pentagon has deep connections to the golden ratio φ = (1+√5)/2 ≈ 1.618. The diagonal of a pentagon divided by its side equals the golden ratio.
The formula involves √(25+10√5) because of the golden ratio relationships in pentagons. This can also be written as (5/4)s²√(5+2√5).
Pentagon-based shapes appear in the Pentagon building, some home plate designs, certain nuts and bolts, and some crystals. They're less common than hexagons because pentagons don't tile a plane.
The apothem is the distance from the center to the midpoint of any side. For a regular pentagon with side s, the apothem ≈ 0.688s.
A hexagonal prism with the same side length has larger volume because hexagons are more circular (higher area-to-perimeter ratio).
Regular pentagons cannot tile a plane without gaps. Only certain irregular pentagons can tile, unlike triangles, squares, and hexagons.
The circumradius (center to vertex) is R = s/(2sin(36°)) ≈ 0.851s.