how to find volume of a cone

Volume of a Cone in Terms of Pi

The volume of a cone in terms of pi is the capacity of the cylinder represented in terms of pi, which signifies the amount of any material it can hold or the amount of any material that can be immersed in it. A cone is a three-dimensional solid object which is having a circle at one end and a pointed end on the other. The pointed end is also known as the vertex of the cone. The material could be any substance or any liquid quantity which can be filled in the cone uniformly. In this section, we will learn about the volume of the cone along with a few solved examples and practice questions.

1.	What is Volume of a Cone in Terms of Pi?
2.	Volume of A Cone in Terms of Pi Formula
3.	How to Calculate Volume of a Cone in Terms of Pi?
4.	FAQs on Volume of a Cone in Terms of Pi

What is the Volume of a Cone in Terms of Pi?

The volume of a cone in terms of pi can be defined as the space occupied by the right circular cone represented as a product of pi. The symbol of the volume is denoted by (V), the volume of the cone is equal to one-third of the product of the area of the circular base and its height. If we correlate the volume of the cone and the volume of the cylinder, the volume of the cone is one-third of the volume of the cylinder of the same radius and height. The unit of volume of the cone in terms of pi is given in terms of cubic units, m³, cm³, in³, or ft³, etc.

Volume of Cone in Terms of Pi Formula

A cone is a three-dimensional solid object which is having a circle at one end and a pointed end on the other. A right circular cone is generated by a revolving right triangle about one of its legs. To calculate the space occupied by a cone in terms of pi, we find the total space occupied by a revolving right triangle about one of its legs. Therefore, the volume of the cone is given by the product of the area of base and height.

Volume of cone is given as a product of (1/3) by base area and height

For any cone whose base radius is 'r', and height is 'h', the volume will be (1/3) times base times its height. Thus, the cone's volume of base radius "r", and height "h" = (1/3) × (area of base) × height of the cone

Since the volume of the cone is equal to one-third of the volume of a cylinder, it can be written as: Volume of cone = ((1/3)r²h)π
Therefore, the volume of a cone, with radius r and height h in terms of pi is ((1/3)r²h)π cubic units.

How to Calculate Volume of a Cone?

The volume of a cone is equal to ((1/3)r²h)π. The following steps are mentioned below by which we can find the volume of a cone.

Step 1: Calculate the height and the radius of the base of the cone.
Step 2: Then find the square of the radius.
Step 3: Find out the product of the π to the square of its radius of the base. We will get the area of the circular base of the cone.
Step 4: Multiply the area of the circular base with the height of the cone.
Step 5: Multiply the result thus obtained with 1/3.
Step 6: Represent the answer in cubic units.

Examples on Volume of a Cone in Terms of Pi

Example 1: The radius of a cone is 3 units. Find the volume of the cone in terms of pi if the height is 10 units.

Solution: Given the length of the radius of a cone, r = 3 units, and height of the cone, h = 10 units

Using the formula for the volume of the cone in terms of pi,
V = ((1/3)r²h)π
⇒ V = (1/3) 3²× 10 × π
⇒ V = π × 3× 10
⇒ V = 30 π

Therefore, the volume of the cone is approx 30π cubic units.
Example 2: Sam has given data about the volume of the cone in terms of pi which is 49π cubic units. Find the radius of the circular base of the cone if the height of the cone is 3 units.

Solution: Given the volume of the cone, V = 49π and height of the cone, h = 3

Using the formula for the volume of the cone in terms of pi, V = ((1/3)r²h)π
⇒ 49π = (1/3) r²× 3 × π
⇒ 49 = r²
⇒ r = 7

So, the radius of the cone is 7 units.

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Practice Questions on Volume of a Cone in Terms of Pi

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FAQs on Volume of a Cone in Terms of Pi

What is the Volume of a Cone in Terms of Pi?

The volume of a cone in terms of pi can be defined as the space occupied by the right circular cone represented in terms of pi. The unit of volume of a cone in terms of pi is given in terms of cubic units where the unit can be m³, cm³, in³, or ft³, etc.

What is the Formula of Volume of a Cone in Terms of Pi?

The volume of a cone in terms of pi is given as, V = (1/3)πr²h, where "V", "r" and "h" are the volume, radius, and h is the height of the cone. A cone is a three-dimensional geometric figure it has a curved surface pointed towards the top and also has a flat surface. The apex is a pointed end of the cone, whereas the flat surface is called the base.

What is the Formula of Volume of a Cone in Terms of Pi and Slant Height?

We use the relation V = (1/3)πr²√(l²−r²) to determine the volume of a cone in terms of pi and slant height where where "V", "r" and "l" are the volume, radius, and l is the slant height of the cone as the value of l can be obtained using the Pythagoras theorem as l = √(r² + h²).

How to Find the Volume of a Cone in Terms of Pi?

We can find the volume of a cone in terms of pi using the below steps:

Step 1: Identify the height and the radius of the base of the given cone.
Step 2: Then find the square of the radius and multiply it to π
Step 3: Multiply this value with the height of the cone.
Step 5: Multiply the obtained answer with 1/3.
Step 6: Write the answer in cubic units.

How to Find the Radius of Cone If the Volume of a Cone in Terms of Pi is Known?

We can find the radius of the cone if the volume of the cone in terms of pi is given using the below steps:

Step 1: Identify the given dimensions of the cone and let the radius of the cone is "r"
Step 2: Square of the radius and multiply it by π.
Step 3: Find the product of the result with the height of the cone.
Step 4: Divide the obtained answer in Step 3 by 3 and solve for "r".
Step 5: Once the equation is solved for "r", represent the obtained answer in units.

What Happens to the Volume of a Cone in Terms of Pi If the Height of Cone is Doubled?

The volume of the cone in terms of pi doubles, if the height of the cone is doubled as V = (1/3)πr²h and the value of "h", will be substituted by "2h" which gives the formula V = (1/3)πr²(2h) = 2 × ((1/3)πr²h) that is twice the original volume of the cone in terms of pi.

What Happens to the Volume of a Cone in Terms of Pi If the Radius of Cone is Tripled?

The volume of the cone in terms of pi becomes nine times its original value, as the radius of the cone is tripled as V = (1/3)πr²h and the value of "r", will be substituted by "3r" which gives the formula V = (1/3)π(3r)²h = 9 × ((1/3)πr²h) that is nine times the original volume of the cone in terms of pi.