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A cone is defined as a three dimensional geometric shape that is obtained by rotating a right angle triangle about one of its bases or perpendicular. It has one curved lateral surface and a circular base. In this article we will explore the formulas to calculate the area and volume of a cone....Read MoreRead Less

A cone is a three-dimensional geometric shape with a circular base that tapers to a point called the ** apex **or

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

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.

**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, \(V~=~\frac{1}{3}~\pi~r^2h\)**

**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**** \(cm^3\)**

Frequently Asked Questions

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.