Elongated Bipyramid Calculator

Name: Elongated Bipyramid: Elongated Triangular, Square and Pentagonal Bipyramid - Geometry Calculator
Author: Jürgen Kummer

Calculations at regular elongated bipyramids. These are three Johnson solids, where on the elongated pyramid another regular pyramid is attached on the other side. There are the elongated triangular bipyramid (J₁₄), the elongated square bipyramid(J₁₅) and the elongated pentagonal bipyramid (J₁₆).
Enter the type of elongated bipyramid and one value and choose the number of decimal places. Then click Calculate.

	Type of elongated bipyramid:			Elongated triangular bipyramid, J₁₄ 9 faces, 15 egdes, 8 vertices Pyramid faces: 6 equilateral triangles, Side surfaces: 3 squares Elongated square bipyramid, J₁₅ 12 faces, 20 egdes, 10 vertices Pyramid faces: 8 equilateral triangles, Side surfaces: 4 squares Elongated pentagonal bipyramid, J₁₆ 15 faces, 25 egdes, 12 vertices Pyramid faces: 10 equilateral triangles, Side surfaces: 5 squares
	Edge length (a):
	Height (h):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to decimal places.

	Formulas: Elongated triangular bipyramid: $h = (\frac{2}{3} \cdot \sqrt{6} + 1) \cdot a$ $A = \frac{3}{2} \cdot (2 + \sqrt{3}) \cdot a^{2}$ $V = \frac{2 \sqrt{2} + 3 \sqrt{3}}{12} \cdot a^{3}$ Elongated square bipyramid: $h = (\frac{2}{\sqrt{2}} + 1) \cdot a$ $A = (4 + 2 \sqrt{3}) \cdot a^{2}$ $V = (1 + \frac{\sqrt{2}}{3}) \cdot a^{3}$ Elongated pentagonal bipyramid: $h = (2 \cdot \sqrt{\frac{5 - \sqrt{5}}{10}} + 1) \cdot a$ $A = [\frac{5}{2} \sqrt{3} + 5] \cdot a^{2}$ $V = [\frac{5 + \sqrt{5}}{12} + \frac{\sqrt{25 + 10 \cdot \sqrt{5}}}{4}] \cdot a^{3}$

Length and height have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The properties of mirror symmetry and rotational symmetry are essentially the same for the three elongated bipyramids as for the two Johnson bipyramids. However, the horizontal plane of symmetry here is midway between and parallel to the two bases, the base doesn't lie within this plane. The elongated square bipyramid has five planes of symmetry, whereas the square bipyramid does not occur among the Johnson bipyramids. The elongated square bipyramid has a rotational symmetry at its vertical axis of rotation at an angle of 90 degrees and multiples thereof. Its horizontal axes of rotation pass through two opposite side vertices or through two opposite side edges.
The elongated square bipyramid is point-symmetric, unlike the elongated triangular bipyramid and the elongated pentagonal bipyramid.

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