Why Total Surface area of Right Circular Cone is πr(l + r)

Total Surface area of Right Circular Cone

  • The cone is the three-dimensional geometrical figure or object which has the base in the form of circle of some radius. It also has height and slant height.
  • And the right circular cone is that cone which is perpendicular to the circular base of the cone.
  • The surface area of curved surface of cone is called as the curved surface area of the cone.

  • If l is the slant height of cone and r is the radius of circular base and h is the height of the cone then the curved surface area of cone is given by,

 

Curved surface area of right circular cone = π*r*l

  • The total surface area of the cone is the sum of curved surface area and the area of the circular base of the right circular cone.

 

Total surface area of right circular cone = π*r*l + π*r2

Total surface area of right circular cone = π*r*(l + r)

Derivation of Curved Surface Area of right Circular Cone:

  • Let us we have taken the cone of base radius r, slant height l, height h and we cut it along the slant height as shown in figure 2.
  • Then the curved surface formed will be as shown in figure 3 in the form of circle. Now, we divide this portion in the form of number of triangles as shown in figure. Here the base of the triangle is curve which acts as a straight line.
  • We divide this circular portion into the number of triangles having bases b1, b2, b3, …and so on and also all triangles are isosceles triangles with two sides having length l, which is the slant height of the cone.

Thus, we can find the curved surface area of right circular cone as bellow:

Curved surface area of right circular cone = sum of all triangles on curved surface

But we know that area of triangle = ½*base*height

Hence,

Curved surface area of cone = ½*b1*l + ½*b2*l + ½*b3*l +…. and so on

 

= ½*l*(b1 + b2 + b3 + …. and so on)

 

But if we add all the bases of triangles then it will be the circumference of circular base of cone = 2*π*r

Hence, (b1 + b2 + b3 + ….) = 2*π*r

 

Thus,

Curved surface area of right circular cone = ½*l*(2πr)

Curved surface area of right circular cone = π*r*l (Hence proved)

Derivation for Total Surface area of Right Circular Cone:

  • The total surface area of the right circular cone is the sum of curved surface area of right circular cone and the surface area of circular base.
  • Let us consider the right circular cone with slant height l, height h and r be the radius of circular base as shown in figure.

  • Then the curved surface area of right circular cone is given by,

Curved surface area of right circular cone = π*r*l

  • And the base of cone is the circular of radius r whose surface area is given by,

Surface area of circular base of cone = π*r2

  • Thus, the total surface area of right circular cone is given by,

Total surface area of right circular cone = curved surface area of cone + surface area of circular base of cone.

= π*r*l + π*r2

= π*r*(l + r)

 

Thus, the total surface area of right circular cone is given by,

Total surface area of right circular cone = π*r*(l + r)

Hence proved.


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