Surface Area of a Pyramid

A pyramid as a geometrical object, does not require a lot of description due to the popularity of the Egyptian pyramids of Giza. They are unique geometrical shapes that are created from a simple combination of triangular and quadrilateral or polygonal faces joined together. In this article, you will find a simple explanation of pyramid surface area calculation. This includes, calculating surface area of any pyramid with square, rectangular and polygonal bases.

Geometrical Facts About Pyramids
A pyramid is a geometrical object called a 'polyhedron', which literally means 'many headed' object. A polyhedron is a math term for any three dimensional geometrical object, that is characterized by completely flat faces and straight edges. Pyramid is made up of a polygon (geometry term for multisided, closed two dimensional object) connected with a point which is known as 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 2n edges. For example, a pyramid with a square base will have n = 4 sides and it has 5 faces, 5 vertices and 8 edges. Another interesting property of a pyramid is that it is self dual. Any geometrical object may have a dual which is another geometrical object which gets embedded inside it in such a way that the surfaces of the object correspond or 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 pyramid itself. Verify it!

Formulas
After that brief sojourn into the interesting properties of a pyramid as a geometrical object, let us have a look at the surface area formulas for a pyramid. The total surface area can be divided into two parts:

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

The lateral surface area belongs to the triangular faces. The calculation of base surface area is simple as it is usually a triangular, square or rectangular base. The unit used for the calculation of surface area is meter² or m². Here are the formulas for surface area of the various types of pyramids.

Triangular Pyramid
The simplest type of pyramid is the regular tetrahedron, with a triangular base and three lateral faces. All its faces are equilateral triangles, which makes the calculation of area quite simple.

Surface Area of Triangular Pyramid = 4 X (Surface Area of Each Equilateral Triangle Face) = 4 X (√3 a² / 4) = √3 a²

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

Square Pyramid
Such a pyramid has 5 vertices and 5 faces. The surface area is split into two parts.

Total Surface Area of a Regular Pyramid (Square Base) = Base Area (Square Area) + Area of 4 Triangular faces = b² + 2bh

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.

Rectangular Pyramid
There is another variety of pyramids, which has a rectangular base. Following is the formula for its surface area:

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 rectangular base respectively, 'h' is the slant height. The perimeter is the sum of the length of sides of any polygon.

The formula for rectangular pyramid's surface area can be generalized for a pyramid with polygonal base as:

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

Using this formula, you can find the surface area of any pyramid.

Geometry is a beautiful subject to 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!