Set of pyramids | |
---|---|
Faces | n triangles, 1 n-gon |
Edges | 2n |
Vertices | n
+ 1 |
Symmetry group | C_{nv} |
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. Pyramids can have from 3 to a virtually unlimited amount of sides.
An n-sided pyramid 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 the polygon, an n-gonal pyramid will has 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 equilateral triangles, and in that case they are Johnson solids.
Star pyramids
Pyramids with regular star polygon bases can also be constructed.
For example the pentagrammic pyramid has a pentagram base and 5 intersecting equilateral triangle sides.
Volume
The volume of a pyramid is $ V=\tfrac{1}{3}Bh $ where B is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. So the volume of a pyramid is the volume of a prism divided by 3.
This can be proven using calculus. By similarity, the dimensions of a cross section parallel to the base increase linearly from the apex to the base. Then, the cross section at any height y is the base scaled by a factor of $ 1-\tfrac{y}{h} $ , where h is the height from the base to the apex. Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is $ \frac{A(h-y)^2}{h^2} $ . The volume is given by the integral
- $ \frac{A}{h^2}\int\limits_0^h (h-y)^2\,dy=\frac{-A}{3h^2}(h-y)^3\bigg|_0^h=\tfrac{1}{3}Ah $
(Trivially, the volume of a square pyramid with an apex half the height of its base can be seen to correspond to 1/6 of a cube formed by fitting 6 such pyramids (in opposite pairs) about a center. Since the "base times height" then corresponds to 1/2 of the cube's volume it is therefore 3 times the volume of the pyramid and the factor of 1/3 follows.)
The volume of a pyramid whose base is a regular n-sided polygon with side length s is therefore:
- $ V=\frac{n}{12}\cot\left(\frac{\pi}{n}\right)s^3 $
- $ V=s^3\frac{n}{(12n-24)\tan\left(\frac{180}{n}\right)} $
- $ V=\frac{\sin\left(\frac{360}{n}\right)}{12}s^3=\frac{s^3}{12\csc\left(\frac{360}{n}\right)} $
The volume of a pyramid whose base is a regular n-sided polygon with radius R is therefore:
- $ V=\frac{n\sin\left(\frac{2\pi}{n}\right)}{6}R^2h=\frac{n}{6\csc\left(\frac{2\pi}{n}\right)}R^2h $
This property can be used to derive the volume for cones, as well. See also cone.
Surface area
The surface area of a pyramid is $ A=B+\sqrt{\frac{P^2h^2+4B^2}{4}} $ where B is the base area, P is the base perimeter.
The surface area of a pyramid whose base is a regular n-sided polygon with side length S is therefore:
- $ A=A_n+nA_3=\left(\frac{n\cot\left(\frac{\pi}{n}\right)}{4}+n\sqrt{\frac{3}{16}}\right)s^2 $
- $ A=\frac{n}{(n-2)\tan\left(\frac{180}{n}\right)}s^2 $
- $ A=\sin\left(\frac{360}{n}\right)s^2 $