A pyramid is a polyhedron with a polygon base of any shape, and all other faces are triangles which have common vertex.



Properties of a Pyramid

  1. Pyramid with a section parallel to the baseThe lateral faces are all triangles meeting at the vertex of the pyramid
  2. The altitude of the pyramid is shortest distance between the vertex and the base. It is the drop distance from the vertex perpendicular to the base.
  3. If a cutting plane parallel to the base will pass through the pyramid, the smaller pyramid thus formed is similar to the original pyramid. By similar solids, $\dfrac{A_b}{A_y} = \dfrac{h^2}{y^2}$.
  4. If two pyramids have equal base area and equal altitude, any section made by a cutting plane parallel to the base are equal. From the figure, if $A_{b1} = A_{b2}$ then $A_{y1} = A_{y2}$.
  5. The pyramid is said to be a right pyramid if the vertex is directly above the centroid of the base, otherwise it is an oblique pyramid


Name of a Pyramid
The name of a pyramid is according to its base. If the base is square, it is called square pyramid, and if the base is pentagon, it is called pentagonal pyramid. Triangular pyramid is also called tetrahedron.

Formulas for Pyramid

Area of the base, Ab
The area of the base is according to the shape of the base polygon. There is no specific formula for this except for regular pyramids.


Lateral Area, AL
The lateral area of the pyramid is equal to the sum of the areas of lateral faces that are triangles. There is no specific formula for this except for right regular pyramids.


Total Area, AT
$A_T = A_b + A_L$


Volume, V
$V = \frac{1}{3}A_b \, h$


The Regular Pyramid

A regular pyramid is one whose base is a regular polygon whose center coincides with the foot of the perpendicular dropped from the vertex to the base.

Regular Polygon and one Lateral Face


Properties of a Regular Pyramid

  1. The edges of a regular pyramid are equal; it is denoted by e.
  2. The lateral faces of a regular pyramid are congruent isosceles triangles (see figure).
  3. The altitudes of the lateral faces of a regular pyramid are equal. It is the slant height of the regular pyramid and is denoted by L.
  4. The altitude of the regular pyramid is perpendicular to the base. It is equal to length of the axis and is denoted by h.
  5. The vertex of regular pyramid is directly above the center of its base when the pyramid is oriented as shown in the figure.
  6. If a cutting plane is passed parallel to the base of regular pyramid, the pyramid cut off is a regular pyramid similar to the original pyramid.


Formula for regular pyramid

Area of the base (note: the base is a regular polygon), Ab
$A_b = \dfrac{n}{2}xr$

$A_b = \dfrac{n}{2}R^2 \sin \theta$

For more information about the two formulas above, see The Regular Polygon.


Area of one Lateral Face, A1
$A_1 = xL$


Lateral Area, AL
$A_L = nA_1$

$A_L = \dfrac{n}{2}xL$

$A_L = \dfrac{PL}{2}$


Total Area, AT
$A_T = A_b + A_L$


Length of lateral edge, e
$e = \sqrt{R^2 + h^2}$


Slant height, L
$L = \sqrt{r^2 + h^2}$

$L = \sqrt{e^2 - (x/2)^2}$


Volume, V
$V = \frac{1}{3}A_b \, h$

$V = \dfrac{n}{6}xr \, h$

$V = \dfrac{n}{6}R^2h \, \sin \theta$


Ab = area of the base (regular polygon)
A1 = area of one lateral face
AL = lateral area
AT = total area
x = length of side of the base
h = altitude of pyramid (this is the length of axis of the pyramid)
L = slant height of pyramid (this is the altitude of triangular face)
P = perimeter of the base
e = length of lateral edge
For x, R, r, n, and θ, see The Regular Polygon.


The Regular Tetrahedron

Regular tetrahedron is one of the regular polyhedrons (Platonic solids). It is a triangular pyramid whose faces are all equilateral triangles.

Regular Tetrahedron. One face of regular tetrahedron.


Properties of a Regular Tetrahedron

  1. There are four faces of regular tetrahedron, all of which are equilateral triangles.
  2. There are a total of 6 edges in regular tetrahedron, all of which are equal in length.
  3. There are four vertices of regular tetrahedron, 3 faces meets at any one vertex.
  4. A regular tetrahedron can circumscribe a sphere that is tangent to all the faces of the tetrahedron.
  5. A regular tetrahedron can be inscribed in a sphere that passes through all the vertices of tetrahedron.
  6. The center of the inscribed sphere, the center of the circumscribing sphere, and the center of the regular tetrahedron itself are coincidence.


Formulas for Regular Tetrahedron

Area of one face, Ab
$A_b = \frac{1}{2}a^2 \sin \theta$

$A_b = \frac{1}{2}a^2 ( \frac{\sqrt{3}}{2} )$

$A_b = \dfrac{a^2 \sqrt{3}}{4}$


Total area, AT
$A_T = 4A_b$

$A_T = 4 ( a^2 \sqrt{3} / 4 )$

$A_T = a^2 \sqrt{3}$


Slant height, L
$L = a \sin 60^\circ$

$L = a ( \frac{\sqrt{3}}{2} )$

$L = \dfrac{a\sqrt{3}}{2}$


Altitude, h
$(\frac{2}{3}L)^2 + h^2 = a^2$

$\frac{4}{9}L^2 + h^2 = a^2$

$\frac{4}{9}(a \sqrt{3} / 2)^2 + h^2 = a^2$

$\frac{4}{9}( \frac{3}{4} a^2 ) + h^2 = a^2$

$\frac{1}{3}a^2 ) + h^2 = a^2$

$h^2 = \frac{2}{3}a^2 )$

$h = a \sqrt{\frac{2}{3}}$

$h = a \sqrt{\frac{2}{3} \times \frac{3}{3}}$

$h = a \sqrt{\frac{6}{9}}$

$h = \dfrac{a\sqrt{6}}{3}$


Volume, V
$V = \frac{1}{3}A_b \, h$

$V = \dfrac{1}{3} \left( \dfrac{a^2 \sqrt{3}}{4} \right) \left( \dfrac{a\sqrt{6}}{3} \right)$

$V = \dfrac{a^3 \sqrt{18}}{36}$

$V = \dfrac{a^3 \sqrt{9(2)}}{3(12)}$

$V = \dfrac{a^3 (3 \sqrt{2})}{3(12)}$

$V = \dfrac{a^3 \sqrt{2}}{12}$