What is the Volume of a Cone?
A cone is a three-dimensional geometric shape with a circular base that tapers to a point called the apex or vertex from a flat base.
The volume of a cone is calculated to decipher the amount of space or capacity it takes up.
In the case of a conical cup, the volume of the cup is determined by the amount of water that fills it.
List Formulas for the Volume of a Cone
The list below gives us the two formulas that are applied to calculate the volume of a cone.
1. V = \(\frac{1}{3}\) B h is the volume of a cone
2. V = \(\frac{1}{3}~\pi r^2\) h is the volume of a cone.
What is the Volume of a Cone Formula?
To calculate the volume of a cone, we need the radius, or the diameter of the circular base or top, and the height of the cone.
The volume of the cone is the product of the area of the base and the height of the cone. The volume of a cone is measured in cubic units.
Mathematically, the volume of a cone can be expressed as
Volume of the cone = \(\frac{1}{3}\) B h
Where,
- B = \(\pi r^2\) is the base area
- h = the height of the cone
Therefore, we can use the formula,
Volume = \(\frac{1}{3}~\pi r^2\)h, to find the volume of a cone.
The formula given above is for both oblique and right cones.
Solved Examples on Cone Volume Formulas
Example 1: Find the volume of the cone if the base area and height are given as 12 sq. ft and 4 ft, respectively.
Solution:
The base area = 12 sq. ft. The height = 4 ft.
V = \(\frac{1}{3}\) B h Write the formula for the volume of a cone
V = \(\frac{1}{3}\) (12)(4) Substitute B with 12, and h with 4
V = 4 × 4 Simplify
= 16 Multiply
The volume of the cone is 16 cubic feet.
Example 2: Find the volume of the given cone.
Solution:
The diameter is 18 meters. So, the radius will be \(\frac{18}{2}\) = 9 meters
V = \(\frac{1}{3}\) B h Write the formula for the volume of a cone
V = \(\frac{1}{3}πr^2 h\) Substitute formula for B
V = \(\frac{1}{3}× π × (9)^2\)×5 Substitute r with 9, and h with 5
V = 135 π Simplify
V = 135 × 3.14 Replace with 3.14
\(\approx\) 423.9 Simplify
The volume of the cone is 423.9 cubic meters.
Example 3: Find the volume of a conical structure in a park if the diameter of its base is 14 meters and the height of the structure is 5 meters.
Solution:
The diameter is 14 meters. So, the radius will be \(\frac{14}{2}\) = 7 meters
V = V = \(\frac{1}{3}\) B h Write the formula for the volume of a cone
V = V = \(\frac{1}{3}π~(7)^2\) 5 Substitute r with 7, and h with 5
V = 245\(\pi\) Simplify
V = 245 × 3.14 Replace with 3.14
\(\approx\) 769.3 Simplify
The volume of the cone is 769.3 cubic meters.
Example 4: Find the volume of the cone that has a height of 6 meters and a base diameter of 16 meters.
Solution:
The diameter of the base is 16 meters. Therefore, the radius of the base will be \(\frac{16}{2}\) = 8 meters.
V = \(\frac{1}{3}π~r^2\)h Formula for the volume of a cone
= \(\frac{1}{3}π~(8)^2\) (6) Replace r with 8, and h with 6
= 128π Simplify
= 128 × 3.14 Replace with 3.14
\(\approx\) 401.92 Simplify
The volume of the cone is 401.92 cubic meters.
Example 5: Find the radius of the oblique cone that has a height of 6 inches and whose volume is 80 cubic inches.
Solution:
V = \(\frac{1}{3}\) B h Write the formula for the volume of a cone
80 = \(\frac{1}{3}~~\pi r^2\) (6) Substitute 6 as the value of h
80 = 2 π \(r^2\) Simplify
\(\frac{80}{2\pi}\) = \(r^2\) Divide each side by 2π
\(\sqrt{\frac{80}{2\pi}}\) = r Take the positive square root of each side
3.56 ≈ r Use a calculator
The radius is about 3.56 inches.
Example 6: The head of a hut is in the shape of a cone. The diameter of the cone is 14 meters and the height is 18 meters. Find the volume of the hut’s head.
Solution:
The diameter of the base of the cone is 14 meters.
Therefore, the radius of the top will be \(\frac{14}{2}\) =7 meters.
V = \(\frac{1}{3}~~\pi r^2\) h Formula for the volume of a cone
= \(\frac{1}{3}~~\pi~(7)^2\) (18) Replace r with 7 and h with 18
= 294 \(\pi\) Simplify
= 294 × 3.14 Replace with 3.14
\(\approx\) 923.16 Simplify
The volume of the hut’s head is 923.16 cubic meters.
Example 7: Find the volume of the birthday hat if its height is 9 inches and the diameter of its top is 4 inches.
Solution:
The diameter of the top of the cone is 4 inches.
Therefore, the radius of the top will be \(\frac{4}{2}\) = 2 inches.
V = \(\frac{1}{3}~~ r^2\) h Formula for the volume of a cone
= \(\frac{1}{3}~~\pi (2)^2\) (9) Replace r with 2, and h with 9
= 12 π Simplify
= 12 × 3.14 Replace with 3.14
\(\approx \) 37.68 Simplify
The volume of the birthday hat is 37.68 cubic inches.
The volume of a cone with the same base radius and height cannot be found directly using the formula base area times h, which is quite intuitive because a cone with the same dimensions will have less volume. Its volume was reduced to one-third of that of the cylinder.
No, not every cone is one-third of a cylinder. A cone has a volume that is one-third that of a cylinder with the same base radius and height.
The surface area of a right circular cone is calculated using the formula
Total Surface Area of a Cone = πr(l+r)
and the formula for the volume of a right circular cone is V = (1/3)πr2h
We can find the radius through the following steps:
- The lateral height should be squared.
- The height should be squared.
- Subtract the squared height from squared lateral height.
- Find the square root of the result from step 3.
- The radius of the cone is the result.
The slant height is defined as the distance between the vertex and any point on the circumference of the base.
A cone is a solid with a circular base and a single vertex.
We multiply the base area (area of a circle: πr2 by the height and then multiply by 1/3 to get the volume. As a result, Voulume of cone = (1/3)πr2h is the formula for the volume of a cone.