Cone Formula

In our daily life, we have seen cone in many different sizes and color. Moreover, it is a geometrical shape that has many uses. Furthermore, the most common example of the cone is an ice cream cone. Besides, in this topic, we will discuss cone, cone formula, its derivation, and solved example.

Cone

Cone is a 3D (3 dimensional) geometric shape. Furthermore, it has a pointy end on one side and a flat surface on the other side. Also, there are two types of cones, namely right circular cone and oblique cone

Right Circular Cone

The right circular cone is a cone that has a line that touches the top point of the cone in a perpendicular. Moreover, this passes through the center of the circle. Furthermore, the apex (top point) of the cone lies just above the center of a circular base. Besides, it is the most common type of geometric cone. For example, ice cream, traffic cones, etc.

Oblique Cone

In this cone, the base and the apex of the cone ate not perpendicular to each other. Moreover, the top point of the cone is not perpendicular with the base. In addition, the vertex of the cone does not lie above the center of the circular base. Usually, these cones are tilted or slant. For example, a geometrical figure.
Most noteworthy, the cone formula for right circular cone and the oblique cone is the same.

Cone Formula

The cone has two formula one for its surface area and one for its volume (because it is a 3D shape). Now, let’s discuss the formula

The surface area of the cone

If we talk about the surface area of the cone then it is the sum of all lateral side and the base of the cone. Moreover, the cone somewhat resembles a pyramid so the formula of the surface area is related.

Cone surface area = \(\pi r s + \pi r^{2}\)

Derivation of the Formula

r = refers to the radius of the circular base
s = refers to the slant height of the cone
\(\pi\) = refers to the value of pi

The volume of the Cone

The cone has a volume which means it is a 3D shape. Also, the curved surface of the cone joins the apex and base of the cone. Moreover, having volume means that it occupies space. Furthermore, we measure it in a cubic meter.

Besides, finding the volume be sure that all the units are in the same unit.

Volume of the cone = \(\frac{1}{3}\pi r^{2}h\)

Derivation of the Formula

\(\pi\) = refers to the value of pi
r = refers to the radius of the circular base
h = is the height of the cone
\(\frac{1}{3}\) = refers to the value of the one-third value

Besides, the volume of the cone and a cylinder are somewhat related to each other. So, do the volume of prism and pyramid. Besides, if the height of the cylinder and cone are equal then the volume of the cylinder will be three times (3×) the volume of the cone.

Solved Example on Cone Formula 

Example 1

Find the surface area of the cone whose radius is 4 cm and slant height is 8 cm?

Solution:

Surface area of a cone = \(\pi r s + \pi r^{2}\)
A = (3.14 × 4 × 8) + (3.14 × 4 × 4)
A = 100.48 + 50.24 = 15.72 \(cm^{2}\)

So, the surface area of the cone of radius 4 cm and slant height 8 cm is 15.72 \(cm^{2}\).

Example 2

Find the volume of the cone whose radius is 8 cm and height is 18 cm?

Solution:

Volume of the cone = \(\frac{1}{3}\pi r^{2}h\)
V = \(\frac{1}{3}\) × 3.14 × 8 × 8 × 18
V = \(\frac{1}{3}\) × 3617.28 = 1206.4 \(cm^{3}\)
The volume of the cone is 1206.4 \(cm^{3}\)