Surface Area of Cone

The surface area of a cone is the total area occupied by the boundary or surface of a cone. It is measured in square units.

A cone is a 3D shape made by turning a triangle around one of its sides. It has a round base at the bottom and a sharp point at the top called the vertex.

A cone has a curved side and a flat base, so it has two surface areas: the curved surface area and the total surface area. 

Cones are classified into two types: right circular cones and oblique cones.

When the tip of the cone is exactly above the center of the base, it forms a neat, even shape called a right circular cone. If the vertex is not above the middle of the base and the cone leans to one side, then it is an oblique cone.

A cone has a curved surface, and it has two types of surface areas: the curved surface area and the total surface area. Look at the cone below to see its surface area, height(h), slant height(l), and radius(r).

A cone has two types of surface areas:

• Curved Surface Area of a Cone
   
• Total Surface Area of a Cone 

Curved Surface Area of a Cone

The area of the curved part of the cone, excluding its base, is known as the curved surface area of a cone. The curved surface area of the cone is called the lateral surface area of a cone. The formula for the CSA (Curved Surface Area) of the cone is given as:
Curved Surface Area  = 𝜋rl square units
Here, r is the radius of the base of a cone.
l is the slanting height of the cone.

Total Surface Area of a Cone

The total area occupied by the cone, including the area of the curved surface and the area of the circular base, is known as the total surface area of the cone. The total surface area of a cone is calculated by using the formula:

Total surface area = πr(r + l) square units
Where r is the radius of the base of the cone.
L is the slant height of the cone.

To find the total surface area of a cone, imagine slicing it from the tip to the edge of the base and unfolding it. When we spread it out flat, the curved surface becomes a part of a circle, called a sector. Let us take a cone with height(h), radius of the base(r), and a slant height(l).

Total surface area of a cone = base area of a cone + curved surface area of a cone

Here, the base area of a cone = 𝜋r²
Curved surface area of a cone = 𝜋rl

Substituting the formulas into the total surface area, 

Total surface area of a cone, T = 𝜋r² + 𝜋rl
Take the common terms out: 
T = 𝜋r(r + l)

Therefore, the total surface area of cone T = 𝜋r(r + l)

The volume of a cone shows how much space is inside it. It tells us how much space is inside the cone or how much it can hold. It is one-third of the volume of a cylinder with the same height and base. The volume of a cone can be found by using the formula:

Volume = ⅓ (𝜋r²h) (cubic unit)

The surface area of a cone helps us understand how much space is covered by the cone. It is used in may fields such as construction, manufacturing, and packaging. Below are some real-life examples where it is applied.

• Food Industry: In making ice cream cones and packaging, surface area is used to figure out how much chocolate or coating is needed to cover the cone and how much material is needed to wrap it. 

• Road Safety: In road safety, traffic cones are used to control traffic. Surface area helps in calculating how much material or paint is needed to make or cover each cone, especially if reflective strips are added.

• Architecture: In designing cone-shaped roofs and tents, surface area helps builders know how much fabric, tiles, or paint is needed to cover the outer surface.