Elongated Pyramid Calculator

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

Calculations with regular elongated pyramids. These are three Johnson solids: the elongated triangular pyramid (J₇) is a regular tetrahedron with a matching right prism attached to one face. Elongated square pyramid (J₈) and elongated pentagonal pyramid (J₉) are the two Johnson pyramids with matching cube, respectively prism attached.
Enter the type of elongated pyramid and one value and choose the number of decimal places. Then click Calculate.

	Type of elongated pyramid:			Elongated triangular pyramid, J₇ 7 faces, 12 egdes, 7 vertices Pyramid faces: 3 equilateral triangles, Side surfaces: 3 squares, Base surface: equilateral triangle Elongated square pyramid, J₈ 9 faces, 16 egdes, 9 vertices Pyramid faces: 4 equilateral triangles, Side surfaces: 4 squares, Base surface: square Elongated pentagonal pyramid, J₉ 11 faces, 20 egdes, 11 vertices Pyramid faces: 5 equilateral triangles, Side surfaces: 5 squares, Base surface: regular pentagon
	Edge length (a):
	Height (h):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to decimal places.

	Formulas: Elongated triangular pyramid: $h = (\frac{\sqrt{6}}{3} + 1) \cdot a$ $A = (3 + \sqrt{3}) \cdot a^{2}$ $V = \frac{\sqrt{2} + 3 \sqrt{3}}{12} \cdot a^{3}$ Elongated square pyramid: $h = (\frac{1}{\sqrt{2}} + 1) \cdot a$ $A = (5 + \sqrt{3}) \cdot a^{2}$ $V = (1 + \frac{\sqrt{2}}{6}) \cdot a^{3}$ Elongated pentagonal pyramid: $h = (\sqrt{\frac{5 - \sqrt{5}}{10}} + 1) \cdot a$ $A = [\frac{\sqrt{25 + 10 \cdot \sqrt{5}}}{4} + \frac{5}{4} \sqrt{3} + 5] \cdot a^{2}$ $V = [\frac{5 + \sqrt{5}}{24} + \frac{\sqrt{25 + 10 \cdot \sqrt{5}}}{4}] \cdot a^{3}$

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.

These shapes resemble houses. With the same edge length, the height decreases as the number of corners increases, while the surface area and the volume increase. The regular, elongated pyramids are mirror-symmetrical with respect to the planes through a vertex of the pyramid's base, the middle of an opposite edge of that base, and the apex, if there are three or five vertices. With a square base, two planes pass through opposite vertices and two pass through the middle of opposite edges, and through the apex. The number of symmetry planes corresponds to the number of vertices at the pyramid's base. These pyramids are also rotationally symmetrical about the vertical axis through the apex at an angle of 360 degrees divided by the number of vertices of the pyramid base, i.e. 120, 90 or 72 degrees respectively.

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