Volume of a Pyramid

Geometry is a branch of Mathematics that teaches us about circles, lines, angles, triangles, pyramids, quadrilaterals, and much more. So, we can say that geometry is all about different types of figures, their construction, and their measurements.  In this article, the highlight is on the volume of a pyramid and its different shapes. We also look at the volume of the pyramid formula and the equations. Pyramids are so-called because of their resemblance to a pyramid structure. It starts with a single point at the top and goes on to become wider at the bottom. The most common shape that you see is the triangular pyramid. A triangular pyramid has four faces hence it is also known as a tetrahedron. See Fig. 1 and Fig. 1a below. 


Definition 

A pyramid is a polyhedron with a base that can be any polygon, with three or more faces that meet at a point called the apex. The triangular sides are called the lateral faces to distinguish them from the base.


Types of Pyramids 

There are different types of pyramids. Mathematicians classify the pyramids according to the shape and form of the base. A pyramid that has a square base is a square-based pyramid. A pyramid with a triangle base is a triangular-based pyramid. No matter what the type of pyramid, they all have one thing in common- their sides are always triangular. 

You all know about the great pyramids of Egypt. They are all square pyramids as their base is square. Ref Fig.2 below. There are different types of pyramid shapes. Some of the common types of pyramids shapes are: 

  1. Triangular pyramid

  2. Square pyramid 

  3. Rectangular pyramid 

  4. Pentagonal pyramid 

  5. Hexagonal pyramid, and others. Ref fig.2 

As you can see in the above-illustrated figure, all the pyramids have a different type of base. The bases of the pyramids range from a square, circle, hexagon (six edges), triangle (three edged), and pentagon (five edges). All the pyramids are named after the shape of its base. 

A square pyramid has a square base, a rectangle pyramid has a rectangle base, a hexagon pyramid has a hexagonal base, and a triangular pyramid has a triangular base. 

It is essential to note that pyramids can additionally be classified in the following types: Oblique pyramid, Right pyramid, Regular pyramid, and Irregular pyramid.  


Volume of a Pyramid 

Volume is a measure that shows how much space an object occupies. For example, two boxes kept together will be twice the volume of a single box. This is due to the fact that two boxes take up double the amount of space.  

The volume of an object has a direct relationship with its shape. Different shapes have different methods to find the volume. For example, in case of a cube, we find the volume by multiplying the three side lengths. If in a cube, the sides’ measure 5cm, then the volume of the cube is 5x5x5 or 125cm3.  

Similarly, we can calculate the volume of a pyramid.  Volume is defined as the total amount of space occupied by a three-dimensional object. Volume is written as V, and is measured in cubic units. 

The formula to calculate volume V = 1/3 x base x height, or V= 1/3 x A X H, where A is the area, H is the height, and V is volume.  Volume is always measured in the following units: in3, ft3, cm3 and m3.  While calculating the volume, always keep the measuring units the same.  


Volume of Pyramid Formula 

Different types of pyramids have different types of base. To work out the volume for triangular, rectangle, square, and hexagon pyramid, we have to make use of different formulas. Let us know more about the formulas and the working. 

Volume of a triangular pyramid –Fig 3

The formula is, 

V= 1/3 x Area of triangular base x Height of pyramid or 

V = 1/3 x (1/2ah) x h (area of triangle is1/2ah)

Therefore, V= 1/6ahH. (Ref. Fig 3)

Volume of square pyramid-Fig 4 

As you know area of a square is A = b2, where b is the length of the side of the square, (Ref. Fig 4). So, Volume of square pyramid is 

V = 1/3 x area of square x Height 

V= 1/3 x b2 x H or V=1/3b2H

For example: Find the volume of a square pyramid. The base is 12 cm each. Height of pyramid is 18 cm. 

Answer. Area of the square is b x b, that is 12 x 12 = 144. 

Substituting the formula, V = 1/3 A H 

    = 1/3 x 144 cm2 x 18 cm 

    = 48 x 18 

Volume = 864 cm3


Volume of a rectangular-based pyramid-Fig 5

= 1/3lwH, as area of rectangle is l x w, and 

For example:  a rectangle pyramid has length of base as 9cm and width as 7cm.  Height of pyramid is 12cm. Find volume. 

Answer. Area of rectangle is = l x w 

Substituting in the formula, V = 1/3 l w H, we get

= 1/3 x 9 x 7 x 12 

Volume = 252 cm3

Volume of a hexagonal pyramid (Ref Fig.6) 

Fig. 6

V= 1/3 x 3√3/2 b2 x H 

V= √3/2 b2

FAQ (Frequently Asked Questions)

Q1. What are the Properties of a Pyramid? 

All pyramids have faces, edges, vertices, surface area, volume, and one apex. These pyramids come together at the apex and at the bottom can be a base of any shape. If their base is square shaped then it is called as a square pyramid. If they have three sides and a base that looks like a triangle then it is called as a tetrahedron. Each of the type of pyramid will have a different volume of pyramid formula. For example, the volume of a tetrahedron will be different from the volume of square pyramid. 

Q2. What are the Properties of a Triangular Pyramid?

The simplest regular pyramid has 4 sides- base plus 3 sides. It is known as tetrahedron- all its triangular sides are congruent.  They posses some unique properties. All triangular-based pyramids have four vertices. Its faces too have some unique properties. It has four triangular faces which start from a common apex at the top. It has six edges overall. If all six edges are of the same length then it is called as a regular tetrahedron. The volume of a triangular pyramid is to factor in the area of the triangular base and multiply it by the height of the pyramid (this height will start from the apex and meet the base at 90° angle). Then, divide this result by three.