# 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.

**Properties of a Regular Pyramid**

- The edges of a regular pyramid are equal; it is denoted by e.
- The lateral faces of a regular pyramid are congruent isosceles triangles (see figure).
- 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.
- The altitude of the regular pyramid is perpendicular to the base. It is equal to length of the axis and is denoted by h.
- The vertex of regular pyramid is directly above the center of its base when the pyramid is oriented as shown in the figure.
- 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**

_{b}

$A_b = \dfrac{n}{2}xr$

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

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

_{1}

$A_1 = xL$

_{L}

$A_L = nA_1$

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

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

_{T}

$A_T = A_b + A_L$

$e = \sqrt{R^2 + h^2}$

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

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

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

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

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

A

_{b}= area of the base (regular polygon)

A

_{1}= area of one lateral face

A

_{L}= lateral area

A

_{T}= 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.

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