Volume of a Cone Formula

Volume of a Cone

The volume of a cone defines the space or the capacity of the cone. A cone is a three-dimensional geometric shape having a circular base that tapers from a flat base to a point called apex or vertex. A cone is formed by a set of line segments, half-lines or lines connecting a common point, the apex, to all the points on a base that is in a plane that does not contain the apex.

A cone can be seen as a set of non-congruent circular disks that are stacked on one another such that the ratio of the radius of adjacent disks remains constant.

A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex.

The volume of a 3 -dimensional solid is the amount of space it occupies.  Volume is measured in cubic units ( in3,ft3,cm3,m3, et cetera).  Be sure that all of the measurements are in the same unit before computing the volume.

The volume V of a cone with radius r is one-third the area of the base B times the height h .

Note : The formula for the volume of an oblique cone is the same as that of a right one.

The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is three times as much as the volume of a cone.

Volume of a Cone Formula

In general, a cone is a pyramid with a circular cross-section. A right cone is a cone with its vertex above the center of the base. It is also called right circular cone. You can easily find out the volume of a cone if you have the measurements of its height and radius and put it into a formula.

Therefore, the volume of a cone formula is given as

The volume of a cone = (1/3) πr²h cubic units

Where,

‘r’ is the base radius of the cone

‘l’ is the slant height of a cone

‘h’ is the height  of the cone

As we can see from the above cone formula, the capacity of a cone is one-third of the capacity of the cylinder. That means if we take 1/3rd of the volume of the cylinder, we get the formula for cone volume.

Note: The formula for the volume of a regular cone or right circular cone and the oblique cone is the same.

Derivation of Cone Volume

You can think of a cone as a triangle which is being rotated about one of its vertices. Now, think of a scenario where we need to calculate the amount of water that can be accommodated in a conical flask. In other words, calculate the capacity of this flask. The capacity of a conical flask is basically equal to the volume of the cone involved. Thus, the volume of a three-dimensional shape is equal to the amount of space occupied by that shape. Let us perform an activity to calculate the volume of a cone.

Take a cylindrical container and a conical flask of the same height and same base radius. Add water to the conical flask such that it is filled to the brim. Start adding this water to the cylindrical container you took. You will notice it doesn’t fill up the container fully. Try repeating this experiment for once more, you will still observe some vacant space in the container. Repeat this experiment once again; you will notice this time the cylindrical container is completely filled. Thus, the volume of a cone is equal to one-third of the volume of a cylinder having the same base radius and height.

Now let us derive its formula. Suppose a cone has a circular base with radius ‘r’ and its height is ‘h’.  The volume of this cone will be equal to one-third of the product of the area of the base and its height. Therefore,

V = 1/3 x Area of Circular Base x Height of the Cone

Since, we know by the formula of area of the circle, the base of the cone has an area (say B) equals to;

B = πr²

Hence, substituting this value we get;

V = 1/3 x πr² x h

Where V is the volume, r is the radius and h is the height.

Solved Examples

Q.1: Calculate the volume if r= 2 cm and h= 5 cm.

Solution:

Given:

r = 2

h= 5

Using the Volume of Cone formula

The volume of a cone = (1/3) πr²h cubic units

V= (1/3) × 3.14 × 2² ×5

V= (1/3) × 3.14 × 4 ×5

V= (1/3) × 3.14 × 20

V = 20.93 cm³

Therefore, the volume of a cone = 20.93 cubic units.

Q.2: If the height of a given cone is 7 cm and the diameter of the circular base is 6 cm. Then find its volume.

Solution: Diameter of the circular base = 6 cm.

So, radius = 6/2 = 3 cm

Height = 7 cm

By the formula of cone volume, we know;

V = 1/3 πr²h

So by putting the values of r and h, we get;

V = 1/3 π 3² 7

Since π = 22/7

Therefore,

V = 1/3 x 22/7 x 3² x 7

V = 66 cu.cm.

Frequently Asked Questions on Volume of Cone

What is the formula for the slant height of a cone?

The slant height of a cone l = √(h² + r²), where h is the height of the cone and r is the radius of the circular base.

What is the formula for the total surface area of a cone?

The total surface area of a cone is given by 𝜋r(l + r) square units, where r is the radius of the circular base and l is the slant height of the cone.

What is the formula for the curved surface area of a cone?

The curved surface area of a cone is given by 𝜋rl square units, where r is the radius of the circular base and l = √(h² + r²) is the slant height of the cone.

How to find the volume of a cone, if its height and diameter are given?

We know that the volume of a cone = (1/3)πr²h cubic units
Since r = d/2, the volume of a cone becomes
V = (1/3)π(d/2)²h cubic units
V = (1/12)πd²h cubic units.
Hence, the formula for the volume of a cone is (1/12)πd²h cubic units, if its height and diameter are given.

What will be the volume of a cone if its radius and height are doubled?

If r = 2r and h = 2h, then the volume of a cone is given as:
Volume of a cone = (1/3)π(2r)²(2h) cubic units
V = (⅓)π(4r²)(2h)
V = (8/3)πr²h
Thus, the volume of a cone becomes (8/3)πr²h, when its radius and height are doubled.

What is the base surface area of a cone?

The base surface area of a cone is a circle, which equals πr².

Solution

From the figure, the radius of the cone is 8

cm and the height is 18

cm.

The formula for the volume of a cone is,

The volume of a cone is defined as the amount of space or capacity a cone occupies. The volume of cone is measured in cubic units like cm³, m³, in³, etc. A cone can be formed by rotating a triangle around any of its vertices. A cone is a solid 3-D shape figure with a circular base. It has a curved surface area. The distance from the base to the vertex is the perpendicular height. A cone can be classified as a right circular cone or an oblique cone. In the right circular cone, a vertex is vertically above the center of the base whereas, in an oblique cone, the vertex of the cone is not vertically above the center of the base.

The volume of a cone formula is given as one-third the product of the area of the circular base and the height of the cone. According to the geometric and mathematical concepts, a cone can be termed as a pyramid with a circular cross-section. By measuring the height and radius of a cone, you can easily find out the volume of a cone. If the radius of the base of the cone is “r” and the height of the cone is “h”, the volume of cone is given as V = (1/3)πr²h.

Volume of Cone With Height and Radius

The formula to calculate the volume of a cone, given the height and its base radius is:
V = (1/3)πr²h cubic units

Volume of Cone With Height and Diameter

The formula to calculate the volume of a cone, given the measure of its height and base diameter is:
V = (1/12)πd²h cubic units

Volume of Cone With Slant Height

By applying Pythagoras theorem on the cone, we can find the relation between volume and slant height of the cone.
We know, h² + r² = L²
⇒ h = √(L² – r²)
where,

• h is the height of the cone,
• r is the radius of the base, and,
• L is the slant height of the cone.

The volume of the cone in terms of slant height can be given as V = (1/3)πr²h = (1/3)πr²√(L² – r²).

Here is an activity that shows how the formula for the volume of a cone is obtained from the volume of a cylinder. Let us take a cylinder of height “h”, base radius “r”, and take 3 cones of height “h”. Fill the cones with water and empty out one cone at a time.

Each cone fills the cylinder to one-third quantity. Hence, such three cones will fill the cylinder. Thus, the volume of a cone is one-third of the volume of the cylinder.
Volume of cone = (1/3) × Volume of cylinder = (1/3) × πr²h = (1/3)πr²h

Given the required parameters, the volume of a cone can be calculated by applying the volume of cone formula. The below-given steps can be followed when either the base radius or the base diameter, height, and slant height of cone are known.

• Step 1: Note down the known parameters, ‘r’ as the radius of the base of cone, ‘d’ as diameter, ‘L’ as slant height, and ‘h’ as the height.
• Step 2: Apply the formula to find the volume of cone,
  Volume of cone using base radius: V = (1/3)πr²h or (1/3)πr²√(L² – r²)
  Volume of cone using base diameter: V = (1/12)πd²h = (1/12)πd²√(L² – r²)
• Step 3: Express the obtained result in cubic units.

Example: Find the volume of a cone whose radius is 3 inches and height is 7 inches. (Use π = 22/7).
Solution: As we know, the volume of the cone is (1/3)πr²h.
Given that: r = 3 inches, h = 7 inches and π = 22/7
Thus, Volume of cone, V = (1/3)πr²h
⇒ V = (1/3) × (22/7) × (3)² × (7) = 22 × 3 = 66 in³
∴ The volume of cone is 66 in³.

Volume of Cone Tips

• The volume of a hemisphere with radius “r” is equal to the volume of a cone having radius “r” and height equal to ‘2r’. Thus, (1/3)πr²(2r) = (2/3)πr³.
• The volume of a cone can be calculated using the diameter, by dividing the diameter by 2 to find the radius, then applying the value into the volume of a cone formula (1/3)πr²h.

Volume of Cone Examples

What is the Volume of a Cone Formula?

What is Surface Area and Volume of a Cone?

How to Calculate Volume of a Cone Using Calculator?

How Do You Find the Volume of Cone with Diameter and Slant Height?

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].

Solved Examples on Volume of a Cone with Diameter

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