A cone has a three-dimensional shape so calculating its volume can seem a little complicated. To help you understand better, in this article we explain what a cone is as well as how to calculate its volume. We detail the steps one by one and the formulas you have to use to calculate the volume of a cone with accurate examples.
What is a cone in math definition?
In geometry, a cone is a three-dimensional solid figure that narrows smoothly from a circular base to a point called the apex or vertex.
The distance from the vertex to the base is the height of the cone and it’s perpendicular to the base. The circular base is measured by the value of the radius or the circumference and the length of the cone from the vertex to any point of the surface area of the base is called the slant height.
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Volume of a Cone: How to calculate it?
Generally, the volume of a three-dimensional solid is the amount of space it occupies and it’s measured in cubic units.
The volume V of a cone with radius r is one third the area of the base B times the height h. The volume of a cone is less than the volume of a cylinder with the same base and height. It’s actually exactly one third of the volume of a cylinder. The formula is as follows :
V = ⅓ (Area of base) x (height)
V = ⅓ πr²h or V = ⅓ Bh, where B = πr²
Finding the volume of a cone is easy to calculate once you know its height and radius. Here’s a tutorial on how to calculate the volume of a cone from its circular base to its height:
1. Find the radius of the base
Either you have the diameter of the base or the circumference. If you’re given the diameter, divide it by 2 to get the radius. If you were, however, given the circumference, divide it by 2π to get the diameter.
2. Calculate the surface area of the circular base
Once you know the diameter, you can calculate the surface area of the base of a cone. Like we mentioned earlier, the base B formula is : B = πr². So square the radius and multiply it by the value of π to find the area of the circular base.
3. Calculate the volume of the cone
Now that you have what you need to calculate the volume of a cone, all you have to do is follow the formula : V = 1/3Bh, where B = πr².
Now, you need to multiply the area of the base B by the height h and then divide the obtained result by 3. Remember to always state the volume in cubic units because you’ve calculated the volume of a three-dimensional space.
Example
Knowing that the height of the cone is h = 18cm and the radius r = 6cm, calculate the volume of the cone shown below.
Solution
So we have the value of both the radius (6cm) and the height (18cm).
The formula for the volume of a cone is :
V = ⅓ πr²h
We can do it all at once, or we can start by calculating the surface area of the base which is :
Base = πr²
= π x (6)²
=π x 36
= 113.09 cm²
Now we can calculate the volume of the cone:
V = ⅓ πr²h
= ⅓ x 113,09 x 18
=678.2
Therefore, the volume of the cone is about 678.2 cubic centimeters.
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