Volume of a Cone

The way I see it, the only difference between a 2-dimensional figure and a 3-dimensional one is that the students need to learn an additional formula: the volume. Every geometry paper comes with a question on how to calculate volume of different geometric shapes.

Volume of a Cone

What is a cone? Simply put, it is a three dimensional triangle, but with a circular base. It differs from the triangular prism in the sense that the triangle prism has a triangular base, and hence, its sloping surface has three well-defined edges. A cone, on the other hand, has a circular base, hence, its slope is smooth and curved, and has no edges. Therefore, while calculating volume, one needs to take into account its circular base.

A cone is one of the most common geometric shapes. And the volume of a shape is basically the amount of any matter that an object of that shape and specifications can hold. It is important to know its volume of given specifications, so that it is possible to know the how much liquid, gas or solid it can hold. Needless to say, the volume should be the same as that of a solid, liquid or a gas contained in it. Suppose you want to make a pop-sickle in a cone-shaped mold, then if you want to know how much liquid it can hold, you will need to know its volume.

The Equation and Calculation

Two things have to be taken into account when you calculate the surface area and volume of a cone. The first, is its circular base, which brings the 'π' and 'r' along with it, and the second is the slant height. The slant height is especially necessary when it comes to calculating the surface area.

The volume can be calculated as:

¹/₃πr²h

where r is the radius of the circular base and h is the height of the cone from the center of the circular base to the top.

The surface area of the curved surface of a cone can be calculated as:

πrs

where r is the radius and s is the slant height of the cone.

The surface area of the base is:

πr²

Adding the two surface areas, we have the total surface area of the cone.

πr (r + s)

where r is the radius of the circular base and s is the slant height of the cone.

In a lot of math homework sums you'll find questions asking you to find the volume of a truncated cone. A truncated cone is one where the conical top is cut off, and you are left with a circular base and a circular top.

The volume of a truncated cone can be calculated by:

¹/₃ πh (R₁² + R₂² + R₁²*R₂²)

where R₁ is the radius of the base and R₂ is the radius of the circular top.

I have always maintained that the chapters on volume and area are the simplest in math as they involve simply substituting the values given for each variable in the formula and calculating it. Hence, it is definitely one of the sections where you can score really well!