What is a Cone?
A cone is a three dimensional solid that has a circular base and a vertex. A cone is a distinctive three dimensional object whose curved and flat surfaces are pointed towards the top. The tip of the cone is called the vertex or apex. And the flat surface is called the base. It also has a curved surface that is pointed towards the vertex. A can be found by rotating a right triangle about the perpendicular height.
There are two types of cones, the first being a right circular cone and the second an oblique cone. If the vertex of the cone lies vertically above the centre of the circular base, then the cone is called a right circular cone. Otherwise, if the cone is inclined or tilted, then the vertex does not lie on the perpendicular line drawn on the center of the base. This type of cone is called the oblique cone.
What is the Volume of a Cone?
The volume of a cone is the capacity or the amount of space occupied by the outer boundary of the cone. If we take the example of an ice-cream cone, the ice cream inside the cone gives us the volume of the cone. It is measured in cubic units, cubic inches, cubic feet or cubic meters. The volume of a cone is one third of the volume of the smallest cylinder that inscribes it, or in simpler terms one third of the smallest cylinder that can submerge a cone.
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Formula for the Volume of a Cone:
The volume of a cone (V) is one third of the product of the area of the base and the height of the cone. Algebraically, the formula for the volume for the cone is,
V = \(\frac{1}{3}Bh\)
Where, “B” is the area of the base of the cylinder and “h” is the height of the cylinder.
We also need to note that, the base of a cone is a circle. So, the area of the base is given by,
Area of circular base = \(\pi r^2\) sq. units
Where, “r” is the radius of the circular base of the cone.
Therefore, the volume of a cone formula is also written as,
V = \(\frac{1}{3}\pi r^2h\)
Where, “r” is the radius of the base and “h” is the height of the cone.
Check out the Volume of a cone calculator
How to Find the Volume of a Cone with Examples
Example 1: Find the volume of the cone of height 9 feet and base diameter 12 feet.
Solution:
The diameter of the base is 12 feet. Therefore the radius of the base will be \(\frac{12}{2}\) = 6 feet.
V = \(\frac{1}{3}\pi r^2h\) [Formula for volume of a cylinder]
= \(\frac{1}{3}(6^2)(9)\) [Replace r with 6 and h with 9]
= 108π [Simplify]
= 108 × 3.14 [replace \(\pi \) with 3.14]
\(\approx \) 339.12 [Simplify]
The volume of the cone is 339.12 cubic feet.
Example 2: Find the radius of the oblique cone of height 9 inches, whose volume is 80 cubic inches. Round your answer to the nearest tenths.
Solution:
V = \(\frac{1}{3}Bh\) [Write the formula for the volume of a cone]
80 = \(\frac{1}{3}\pi^2(9)\) [Substitute 9 as the value of h]
80 = \(3\pi r^2\) [Simplify]
\(\frac{80}{3\pi} = r^2\) [Divide each side by 3π]
\(\sqrt{\frac{80}{3\pi}}\) = r [Take positive square root of each side]
2.9 ≈ r [Use a calculator]
The radius is about 2.9 cubic inches.
Example 3: Find the volume of the ice-cream as shown in the image.
Solution:
The diameter of the top of the cone is 2 inches.
Therefore, the radius of the top will be \(\frac{2}{2}\) = 1 inch.
V = \(\frac{1}{3}\pi r^2h\) [Formula for volume of a cylinder]
= \(\frac{1}{3}\pi(1^2)(6)\) [Replace r with 1 and h with 6]
= \(2\pi\) [Simplify]
= 2 × 3.14 [replace with 3.14]
\(\approx\) 6.28 [Simplify]
The volume of the ice cream cone is 6.28 cubic inches.
Example 4: The head of a rocket is in the shape of a cone. The diameter of the cone is 15 feet and height is 30 feet. Find the volume of the rocket head.
Solution:
The diameter of the cone base is 30 feet.
Therefore, the radius of the top will be \(\frac{30}{2}\) = 15 feet.
V = \(\frac{1}{3}\pi r^2h\) [Formula for volume of a cylinder]
= \(\frac{1}{3}\pi (15)^2(15)\) [Replace r with 15 and h with 15]
= 1125 [Simplify]
= 1125 × 3.14 [replace with 3.14]
\(\approx \) 3532.5 [Simplify]
The volume of the rocket head is 3532.5 cubic feet.
The volume of a cone with the height and radius is calculated by applying the formula,
Volume of a cone V = (1/3)πr2h cubic units.
In this formula , “r” is the radius of the base, and “h” is the height of the cone.
The volume of a cone formula with height and diameter is,
V = (1/12)πd2h
With regards to this formula, “d” is the diameter of the circular base of the cone and “h” is height of the cone.
A cone has same volume as one third of a cylinder when their base and height is same.