Pyramid

Carefully observe the solid objects in the above figure. Note that their faces are polygons. Of these faces, the horizontal face at the bottom is called the base. All the faces, except the base are of triangular shape. The common point of these triangular faces is called the "apex". A solid object with these properties is called a "pyramid". Note that the bases of the Pyramids shown above are respectively, the shape of a quadrilateral, a pentagon and a hexagon.

Right pyramid with a square base

The base of the pyramid in the figure is a square. All the remaining faces are triangular in shape. If the line segment connecting the apex and the midpoint of the square base (that is the intersection point of the two diagonals) is perpendicular to the base, then such a pyramid is called a "square based right pyramid". The length of the line segment connecting the apex and the midpoint of the base is called the perpendicular height (or simply the height) of the pyramid. The edges of the triangular faces which are not common to the base are called slant edges. In this lesson, we will only consider finding the surface area of square based right pyramids.

Note: A tetrahedron can also be considered as a pyramid. All the faces of a tetrahedron are triangular in shape. Any one of the faces can be taken as the base. The concept of —right pyramid˜ can be defined even when the base is not a square. For example, we can define a right pyramid when the base of a pyramid is a regular polygon, as follows. First note that all the axes of symmetry of a regular polygon pass through a common point, which is called the centroid of the regular polygon. A pyramid, having a base which is a regular polygon, is called a right pyramid, if the line segment connecting the apex and the centroid of the base is perpendicular to the base. If you study mathematics further, you will learn how to define the centroid, even when the base is not a regular polygon.

An important property of a square based right pyramid is that all its triangular faces are congruent to each other. Therefore, all the triangular faces have the same area. Moreover, note that each triangular face is an isosceles or an equilateral triangle, with one side a side of the square base and the other two sides equal in length.

Surface area of a square based right pyramid

To find the total surface area of a square based right pyramid we need to add the areas of the base and the four triangular faces. Suppose the length of a side of the square base is —a˜ and the perpendicular height of a triangular face is "l".