Omkar Phatak
May 4, 2019

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A pyramid is quite an intriguing geometrical object, with many interesting properties. In this post, we present the surface area formula for it.

A pyramid as a geometrical object, does not require an introduction due to the popularity of the Egyptian pyramids of Giza. They are unique shapes, created from a simple combination of triangular and quadrilateral or polygonal faces joined together.

A pyramid is a '*polyhedron*', which literally means 'many headed' object. A polyhedron is any three-dimensional object, that is characterized by completely flat faces and straight edges.

A pyramid is made up of a polygon (*a geometry term for multi-sided, closed two-dimensional object*), connected with a point, which is known as the apex. Every side of the polygonal base, connected with the apex point, forms a triangular surface.

If a pyramid's base has '*n*' sides, then it has '*n + 1*' faces, '*n + 1*' vertices, and '2*n*' edges. For example, a pyramid with a square base will have *n* = 4 sides, 5 faces, 5 vertices, and 8 edges. Another interesting property is its self-dual nature.

Any geometrical object may have a dual, which is another object that gets embedded inside it, in such a way, that the surfaces of the object touch the vertices of its dual. The dual of a polyhedron, like any n-sided pyramid, is the same object. That is, a dual of a pyramid is the same object. Verify it.

After that brief sojourn into the appreciation of geometrical beauty, let us look at the surface area formulas.

*Total Surface Area of a Pyramid = Base Surface Area + Lateral Surface Area*

The lateral surface area belongs to the triangular faces. The calculation for the base is simple, as it is usually a triangular, square, or a rectangular base. The unit used for the calculation is meter^{2} or m^{2}. Here are the formulas for various types of pyramids.

The simplest type of pyramid is the regular tetrahedron, with a triangular base and three lateral faces. All its faces are equilateral triangles, which simplifies calculation.

*Surface Area of Triangular Pyramid = 4 x (Surface Area of Each Equilateral Triangle Face) = 4 x (√3 a*^{2} / 4) = √3 a^{2}

where 'a' is the length of every one of the six edges. Just plug in the value of edge length, to get the answer.

Such a pyramid has 5 vertices and 5 faces.

where 'b' is the base length and 'h' is the slant height of each triangular face. Knowing the value of 'b' and 'h', the calculation is quite straightforward.

There is another variety of pyramids, which has a rectangular base. Following is the formula:

*Surface Area of a Rectangular Pyramid = lb + bh + lh = lb + h (b+l) = Area of Rectangular Base + ½ x h x (Perimeter of Rectangular Base)*

Here, 'l' and 'b' are the length and breadth of the rectangular base respectively, while 'h' is the slant height. The perimeter is the sum of the length of sides of any polygon.

The previous formula can be generalized for a pyramid with a polygonal base as:

*Surface Area of a Pyramid with Polygonal Base = Base Surface Area + [½ x (Base Perimeter) x Slant Height)]*

Geometry is a beautiful subject of study. Arithmetic is often given more importance in mathematics, compared to geometry, but the latter is considerably more interesting, as you will discover, when you delve deeper.