What is a Cone?
A cone is a three-dimensional geometric shape with a circular base that tapers to a point called the apex or vertex.
The distance between the center of the circular base and vertex is known as the height of the cone. The distance between the vertex and any point on the circumference of the base is known as the slant height of the cone. The radius of the circular base is considered as the radius of the cone itself. The height, slant height and radius of the cone are denoted by ‘h’, ‘l’ and ‘r’, respectively.
In our daily life, we come across a number of objects that are in the shape of a cone, for instance, birthday hats, ice-cream cones, and rocket heads.
Formulas Related to the Cone
- Formula for the Slant Height of a Cone
Apply the Pythagorean theorem to the \(\triangle~ABC\).
\(l^2~=~h^2~+~r^2\) Pythagorean theorem
\(l~=~\sqrt{h^2~+~r^2}\) Taking square root on both sides
- Formula for the Curved Surface Area of a Cone
Curved surface area of cone (CSA) = \(\pi~rl\)
Since slant height, \(l~=~\sqrt{h^2~+~r^2}\)
Curved surface area of cone (CSA) = \(\pi~r\sqrt{h^2~+~r^2}\)
Where \(\pi\) is the mathematical constant whose value is \(\frac{22}{7}\) or 3.14.
- Formula for the Total Surface Area of a Cone
The total surface area (TSA) of the cone is the sum of curved surface area and the area of the circular base.
Total Surface Area of Cone (TSA) = \(\pi~rl~+~\pi~r^2~=~\pi~r(l~+~r)\)
- Formula for the Volume of a Cone
The amount of space that has been occupied by the cone is known as volume of cone.
Volume of cone, \(V~=~\frac{1}{3}~Bh\)
Where B is the area of the base.
Since the base is circular with radius \(r\), \(B~=~\pi~r^2\), so,
Volume of a cone, \(V~=~\frac{1}{3}~\pi~r^2h\)
Solved Examples
Example 1: Find the slant height of a cone with a radius of 6 cm and a height of 8 cm.
Solution:
\(l~=~\sqrt{h^2~+~r^2}\) Formula for the slant height of a cone
\(l~=~\sqrt{8^2~+~6^2}\) Substitute the values
\(l~=~\sqrt{64~+~36}\) Square of 8 and 6
\(l~=~\sqrt{100}\) Add
\(l~=10\) Positive square root of 100
Therefore, the slant height of the cone is 10 cm.
Example 2: Find the radius of a cone whose curved surface area (CSA) is 2640 \(cm^2\) and its slant height is 84 cm. (Use \(\pi~=~\frac{22}{7}\))
Solution:
\(CSA~=~\pi~rl\) Formula for the CSA of a cone
\(2640~=~\frac{22}{7}~\times~r~\times~84\) Substitute the given values
\(r=~\frac{2640~\times~7}{22~\times~84}\) Solve for \(r\)
\(r=~10\) Simplify
So, the radius of the cone is 10 cm.
Example 3: Sam bought an ice cream cone. He wonders how much ice cream can be filled in the cone. Can you help him find the quantity of ice-cream needed to fill the cone that has a height of 14 cm and a radius of 6 cm.(Use \(\pi~=~\frac{22}{7}\))
Solution:
As stated in the question,
The height of the cone = 14 cm
The radius of the ice cream cone = 6 cm.
The quantity of ice-cream that can be filled in the cone is determined by calculating the volume of the cone.
\(V=~\frac{1}{3}~\pi~r^2h\) Formula for volume of cone
\(V=~\frac{1}{3}~\times~\frac{22}{7}~\times~6^2~\times~14\) Substitute the given values
\(V=~528\) Simplify
So, the quantity of ice cream that can be filled in the cone is 528 \(cm^3\)
The distance between the center of the circular base and the vertex is known as the height of the cone, and the distance between the vertex and any point on the circumference of the base is known as the slant height of the cone.
The curved surface area (CSA) of a cone is the area that is the area of its lateral face.
A cone has two faces. The first is the circular base, and the second is the curved face of the cone.