A pyramid is a 3-D shape, where the base is a square or triangle. However, the logic can be expanded to any many-sided polygonal base where the sides meet at a single apex point: The top of the pyramid. The volume is calculated using the area at the base of the pyramid and the vertical height to the apex.
In general terms, the formula for the volume of a pyramid is given as follows:
V = 1/3 x Area of the base x vertical height
To calculate the volume of a pyramid, one needs to know the area of the base.
Calculating area of a polygon
Polygons are many-sided shapes. These can be regular or irregular. Regular, in this case, means that the sides are the same length. As such, a rectangle is an irregular polygon.
For a regular polygon, we can calculate the area using an apothem. An apothem is a line segment which joins the polygon’s center to the mid-point of any side. The apothem must be perpendicular to that side.
From this, we get that
A= 1/(2 ) pa
Where p is the perimeter or the sum of all the sides, and a is the length of the apothem. This can get a little complex when you don’t have the apothem, which is why you should use our calculator.
To calculate the area of irregular polygons, one would divide the shape into triangles by drawing all diagonals from one of the vertices, or corners to the others, or by connecting vertices to make bigger shapes. This takes some manual calculation!
There are easier ways of calculating areas for common shapes, for example, a triangle’s area is half its base, multiplied by its height:
A of a triangle= 1/2 b ×h
And the area of a quadrilateral (four-sided shape) is the length times the breadth:
A of a quadrilaterial= l ×b
Finding the volume of a pyramid with a triangular base
The area of a triangle is found by calculating the area of the triangle as shown above and multiplying this by the one-third times the vertical height of the pyramid
V=1/3 (1/2 b x h) x H
In this case h = the 2D height of the triangle base, and H = the vertical height of the pyramid.
Finding the volume of a pyramid with a square base
We know that the area of a quadrilateral, or any four-sided shape is the product of the length and breadth.
A=l × b
For a square, the sides are equal. Therefore, the equation for the volume of a square based pyramid is:
V=1/3 l^2 x h
Findng tolume of a pyramid with a base of more than four sides
This calculator enables you to calculate the volume of a 3-D pyramid for any shaped base, as long as the base shape is a regular polygon, where all sides are the same length.
A pentagon is a 5-sided shape
A hexgon is a 6-sided shape
A heptagon is a 7-sided shape
An Octogon is an 8-sided shape
A Nonagon is a 9-sided shape
And a Decagon is a 10-sided shape
This calculator goes on to calculate the volume for a pyramid with an 11-sided base (hendecagon), as 12-sided base (dodecagon), a 20-sided base (icosagon), a 50-sided base (pentacontagon) and a 100-sided base (hectogon).
Using the Calculator
To calculate any of these volumes, you will need to enter the side length (remember all sides are equal), the vertical height, and the shape of the base. The calculator does the rest!
If you need to calculate for an irregular base (even a rectangle), use the section for base area known, and begin by calculating the area of the rectangle. You can use this function for any irregular base as long as you calculate the base area first.