A done is a 3d shape that tapers from above and has a flattened base. The base of the cone is called an apex or vertex. The number of units that will be covered by the apex or the vertex of a cone is known as the total surface area of a cone.

The total portion or area covered by the surface of the boundary is known as The total surface area of the cone. The total surface area of cone is measured in square units. Attaching many triangles and rotating them around an axis will give you the figure of a cone. 

The surface area cone formula 

If the radius of a cone is “r” and the slant height is “l,” then the formula for the surface area of a cone is given as:-

                  T=πr(r+l)

Derivation of the total surface area of a cone

To find the surface area of the cone, first, we’ll have to calculate the radius of the circular portion of the cone and then find the area of the circle. The area of the circle will be 3.14 times the radius squared (π²). Now to find the area of the cone, we must measure the slant height of the cone. Now use the measurement of the side and find the area of the cone. 

Taking the cone height as “h,” radius as “r,” and slant height as “l.” To determine the surface area of a cone, we have to cut open the cone from the centre. 

The curved surface area can be found by finding the area of the sector by using the formula, 

          

                  S = πrl units^2.

The total surface area of a cone = area of the base of the cone + curved surface area 

Total surface area = πr^2 + πrl = πr (r + l).

total surface area of cone = T = πr (r + l) units^2

The total surface area of a right circular cone

In the right circular cone, the axis is perpendicular to the plane of the base. When a right angle triangle is revolved above one of its legs. The base is in the shape of a circle. There are three elements present in the right circular cone 

Radius

Height 

Slant height

Radius is the length between the centre of the circle base to any point at the circumference of the base. 

Height is the distance from the apex to the centre of the circle base.

The slant height is the distance between the apex and any point on the circumference. 

The total surface area of a right circular cone formula. 

The formula for the total surface area of a right circular cone can be given as:-

            Π(r+l) r

Where r is the radius of the cone, and l is the slant height of the cone. 

Properties of a cone:

There are some properties of the cone that make it separate from other shapes, and they are as follows:-

  • Its circular base.
  • The slant height is the area from the vertex to the edge of the circular base.
  • If a right-angle triangle is circled with a perpendicular side as the axis of the cycle, it will form a right circular cone.

The main properties of a cone are as follows: 

  • One side of the face is circular.
  • The cone has zero corners.

How many types of Cones are there?

A cone is of two types, depending on the place of the vertex:  Denote

  • A circular cone is one whose apex is positioned perpendicular to the base, which explains that the perpendicular line slides precisely on the centre of the base of the cone. The letters ‘h’ denote the cone’s height, and ‘r’ is the radius of the cone.
  • Let us see the other type of cone, so if the arrangement of the vertex is nowhere near the centre of the base of the cone, then it is known as an oblique cone.

Fun facts about cones,

  • A Cone is not just a shape; there is a photoreceptor situated in the retina of the eyes, which helps in better vision.
  • The word ‘Cone’ is the Greek word ‘konos,’ which simply means peak or a wedge.

Conclusion

The total surface area of a cone is the total portion or area covered by the cone’s surface. The total surface area of the cone is measured in square units. Joining many triangles and rotating them around an axis will give you a cone shape. A cone is a 3D shape of an object with a flattened base and a narrow top. The distance from the top of the cone to the cone’s centre is called the cone’s height. The total surface area of a cone can be calculated by  the areas of the base of the cone + curved surface area.