Volume of a Cone with Diameter

In this article, we will learn how to find the volume of a cone with a diameter. First, let us recall what is a cone. A cone is a three-dimensional shape with two faces, the base, and the curved surface. The base is a flat face which is a circle. The curved surface is formed by the set of all line segments that are drawn from each point on the circumference of the base and they all meet at one point, which is called "apex" (or) "vertex" of the cone.

Calculating the volume of a cone with diameter gives us the amount of space that is inside the cone with the help of diameter. It is measured in cubic units (like m³, cm³, in³, etc). The volume of a cone can be expressed in terms of radius and hence it can be expressed in terms of diameter also. Hence the volume of a cone is found with the diameter also. Let us learn the volume of a cone with diameter along with formula, solved examples, and a few practice questions.

The volume of a cone is the total space occupied by the object in a three-dimensional plane. This volume can be expressed in terms of the diameter of the base of the cone. We know that a cone and a cylinder of the same radius and height are connected in terms of volume. The volume of a cone is one-third of the volume of a cylinder whose radius and height are the same as that of the cylinder. You can try an experiment to understand this. Take a conical flask and a cylindrical flask of the same radius and height. Try to fill the cylindrical flask with water by pouring water into it using the conical flask. See how many times you have to use the fully filled conical flask to fill the cylindrical flask completely. Yes, it will be exactly 3 times. i.e., the volume of the cylinder is exactly three times that of the cone.

We know that the volume of a cylinder of radius 'r' and height 'h' is πr²h. In the earlier section, we have learned that the volume of a cone is one-third of that of a cylinder of the same base radius and height. So the volume of a cone of radius 'r' and height 'h' is ⅓ πr²h. But this is the formula for finding the volume of a cone with radius and height given. But what if we are given with diameter? We just make it half to find the radius and then we use the same formula as mentioned earlier to find the volume of the cone.

If 'd' is the diameter of a cone, then its radius is, r = d/2. Substituting this in the above formula,

The volume of the cone = (1/3) π(d/2)²h = (1/3)(1/4) πd²h = (1/12) πd²h.

Thus, the volume of a cone with diameter = (1/12) πd²h.

There are two methods to find the volume of a cone with diameter. Let us consider a cone whose base diameter is 'd' and height is 'h'.

• Method 1:
  Directly substitute 'd' and 'h' values in the formula of the volume of a cone in terms of diameter. i.e.,
  volume = (1/12) πd²h.
• Method 2:
  Find the value of the radius, 'r' using r = d/2, and use the general formula to find the volume of a cone. i.e.,
  volume = ⅓ πr²h.

Here, π is a constant whose value is approximately 3.141592...

Note: Here is the relation among the radius, diameter, and slant height of a cone. This may be helpful when we are asked to find the volume of a cone given the slant height.

√(r² + h²) = √[(d²/4) + h²].

What Is the Volume of a Cone Using the Diameter?

The volume of cone can be calculated using the height and diameter. The base of a cone is a circle and the diameter of this circular base is also known as the diameter of the cone. The formula to find volume of a cone with diameter is given as, V = (1/12) πd²h
where,

• d = Diameter of cone
• h = Height of cone

How Do You Find the Diameter of a Cone With the Volume and Height?

To find the diameter (d) of a cone of volume V and height h, we just substitute all of them in the formula V = (1/12) πd²h and solve for d.

How Do You Find the Volume of a Cone With Diameter and Height?

To find the volume of a cone with diameter d and height h, we have two ways.

1. Finding radius r by making the diameter half and plugging it into the formula, volume = ⅓ πr²h.
2. Substituting d and h directly into the formula, V = (1/12) πd²h.

How To Find the Volume of a Cone With the Diameter and Slant Height?

The relation between the diameter d, height h, and the slant height l of a cone is, l = √[(d²/4)+h²]. We first use this formula, solve for h, and then we substitute d and h into the volume formula (1/12) πd²h.

How Do You Find the Volume of a Cone With a Base Diameter?

If the base diameter (d) and the height (h) of a cone are given, we can find its volume by using the formula (1/12) πd²h. Alternatively, we can find the radius (r) of the cone by making the diameter in half and then apply the volume of a cone formula ⅓ πr²h.

What Is the Formula to Find the Volume of a Cone With Diameter?

The formula to calculate the volume of a cone using the given diameter and height is given as, V = (1/12) πd²h, where, 'd' is diameter of cone, and 'h' = height of cone.