Pyramid (geometry)

Set of pyramids

Faces	n triangles, 1 n-gon
Edges	2n
Vertices	n + 1
Symmetry group	C_nv, [n], (*nn), order 2n
Rotation group	C_n, [n]⁺, (nn), order n
Dual polyhedron	Self-dual
Properties	convex

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base.

A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon and the apex is above the center of the polygon, an n-gonal pyramid will have C_nv symmetry.

Pyramids are a subclass of the prismatoids.

Pyramids with regular polygon faces

The regular tetrahedron, one of the Platonic solids, is a triangular pyramid all of whose faces are equilateral triangles. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of regular convex polygons, in which case they are Johnson solids.

Tetrahedron	Square pyramid	Pentagonal pyramid

Star pyramids

Pyramids with regular star polygon bases are called star pyramids.^[1] For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides.

Volume

The volume of a pyramid (also any cone) is $\scriptstyle {V=}{\tfrac {1}{3}}\scriptstyle {bh}$ where b is the area of the base and h the height from the base to the apex. This works for any polygon, regular or non-regular, and any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. In 499 AD Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.6).^[2]

The formula can be formally proved using calculus: By similarity, the linear dimensions of a cross section parallel to the base increase linearly from the apex to the base. The scaling factor (proportionality factor) is $\scriptstyle {1-}{\tfrac {y}{h}}$ , or $\scriptstyle {\tfrac {h-y}{h}}$ , where h is the height and y is the perpendicular distance from the plane of the base to the cross-section. Since the area of any cross section is proportional to the square of the shape's scaling factor, the area of a cross section at height y is B× ${\tfrac {(h-y)^{2}}{h^{2}}}$ , or since both b and h are constants ${\tfrac {b}{h^{2}}}$ $\scriptstyle (h-y)^{2}$ . The volume is given by the integral

{\frac {b}{h^{2}}}\int _{0}^{h}(h-y)^{2}\,dy={\frac {-b}{3h^{2}}}(h-y)^{3}{\bigg |}_{0}^{h}={\tfrac {1}{3}}bh.

The same equation, $\scriptstyle {V=}{\tfrac {1}{3}}\scriptstyle {bh}$ , also holds for cones with any base. This can be proven by an argument similar to the one above; see volume of a cone.

The volume can also be calculated without knowing b, the area of the base. The volume of a pyramid whose base is an n-sided regular polygon with side length s and whose height is h is therefore:

V={\frac {n}{12}}hs^{2}\cot {\frac {\pi }{n}}.

Surface area

The surface area of a pyramid is $\scriptstyle {A=B+}{\tfrac {PL}{2}}$ where B is the base area, P is the base perimeter and L is the slant height $\scriptstyle {L=}{\sqrt {\scriptstyle {h^{2}+r^{2}}}}$ where h is the pyramid altitude and r is the inradius of the base.

References

^ Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 50, ISBN 978-0-521-09859-5.
^ Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.64, ISBN 978-81-7434-480-9

External links

Weisstein, Eric W. "Pyramid". MathWorld.
Olshevsky, George. "Pyramid". Glossary for Hyperspace. Archived from the original on 4 February 2007.
The Uniform Polyhedra

[1] Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 50, ISBN 978-0-521-09859-5.

[2] Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.64, ISBN 978-81-7434-480-9

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