 # Pyramid

## Pyramid Definition

In geometry, a pyramid can be defined as a polyhedron. It has a base and three or more triangular faces meeting at a point above the base (the apex). Each base, edge and apex of a pyramid form a triangle which is known as a lateral face. It can be said that a pyramid shape is a conic solid with a polygonal base; they are self-dual.  A pyramid with an n-sided base includes n + 1 vertices, n + 1 faces, and 2n edges. In a right pyramid, the apex is directly above the centroid of its base. And the non-right pyramid is known as oblique pyramids.

A regular pyramid is formed of a regular polygon base and is usually implied to be a right pyramid. A triangle-based pyramid is often called a tetrahedron. A pyramid is called acute if its apex is above the interior of the base but if its apex is above the exterior of the base, it is called an obtuse pyramid. Whereas in the case of a  right-angled pyramid, its apex is above an edge or vertex of the base.

### Different Types of Pyramids

Given below are the different types of pyramids:

1. Triangular Pyramid: A pyramid with a triangular base is called a triangular pyramid.

2. Square and Rectangular Pyramid: A square or rectangular pyramid is a three-dimensional geometric figure. It has a square or rectangle base and four triangular sides that meet at one point. for example, the “Great Pyramid of Giza”.

3. Pentagon Pyramid: A pentagonal pyramid has a pentagonal base upon which stands five triangular faces that connect at a point (the vertex).

4. Right  Pyramid:  In a right pyramid the apex lies directly above the centroid of the base. Thus a regular pyramid is a special case of a right pyramid. It helps us to know where the top (apex) of the pyramid is. It can be said that a right pyramid has its apex directly above the centre of the base, it has four sides facing the triangle, it has a square base and it has  5 Vertices (corner points) and It has 8 Edges.

5. Oblique Pyramid: In an Oblique Pyramid, the apex is not over the midpoint or centre of its base. For an irregular pyramid, the base has no defined centre and therefore the difference between right and oblique makes no sense. The four faces of an oblique pyramid are Triangles whereas the base is always a square. It has 5 Vertices (corner points) and 8 Edges.

6. Regular Pyramid: Out of all the different types of Pyramids, regular pyramids are found in abundance. The base of a regular pyramid is always a regular polygon with equal length of lateral edges.

7. Irregular Pyramid: The base of an irregular pyramid is always an irregular polygon.

### The Surface Area of a Pyramid

The surface area of a pyramid can be explained as the number of square units required to cover the entire surface of the pyramid.The total surface area of the pyramid can also be explained as the sum of the area occupied by each surface of a pyramid.

It is easy to find the surface area of the right pyramid as all the surfaces are the same. Thus, we simply find the area of one of the surfaces.  By adding the area of the base we can get the total surface area.

### Total Surface Area

According to Pyramid math the total surface area of a Pyramid can be shown as:

area = ${b^2}$ + 4$\frac{bh}{2}$

Here,

b = side length of the base

h = slant height.

The given equation can also be simplified as :

area = ${b^2}$ + 2bh

The surface area of a regular polygon is easy to find whereas there is no straightforward formula to find the area of an irregular pyramid as each triangular faces of an irregular pyramid is of different shapes and sizes. Given below are some of the ways to find the area of an irregular pyramid.

1. Pyramids can be broken into different triangles. The surface area of these triangles can be calculated separately and added. The figure given below is of a pyramid broken into different triangles.

2. A pyramid can also be represented as a regular hexagon with a missing triangular piece. If the surface area of the missing triangle can be calculated then we can simply find the area of a regular hexagon and subtract the area of the missing triangle from it.

3. A pyramid can also be broken into two parallelograms. Thus, the total surface area can be calculated by adding the areas of two parallelograms. The image given below is of a pyramid divided into two parallelograms by red dotted lines.

4. If the coordinates and vertices of the pyramid are known then we can calculate the area by using the manual method or computer algorithm.

### The Volume of a Pyramid

Volume of a pyramid can be explained as the number of cubic units can exactly fill a pyramid.

The volume of a pyramid can be calculated as one-third of the base area times the perpendicular height. The formula to calculate the volume of the pyramid is given below:

volume = $\frac{1}{3}$ bh

Here,

b = area of the base of the pyramid

h = height ( the vertical distance between apex and base.of a pyramid).

What is Oblique Pyramid? How is the volume of an oblique pyramid calculated?

The word oblique means ‘lean over’. An oblique pyramid is a pyramid whose apex is not exactly over the centre of the base. An oblique pyramid has 4 triangular faces having a square base, 5 vertices and 8 edges. The volume of an oblique pyramid can be found by the same formula which is used to find the volume of other pyramids. The formula to calculate the volume of a pyramid of height (h) and base (b) is:

Volume = 1/3   (bh)

How to find the unknown height of a pyramid?

Let us suppose a pyramid of a base and diagonal is taken. If the height of the pyramid is not known then we can use Pythagora's theorem to find the unknown height of the pyramid such that base is half of ‘a’.

Therefore according to Pythagoras formula,

(Hypotenuse)2= (Base)+ (Height)

(l)2= (a/2)2 + (h)2