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Surface Area of a Cone

Surface area of a cone_1Like a cylinder, a cone is a special type of solid since it has a curved surface. The surface area of a cone is equal to the area of the curved surface plus the area of the circular base. It is given by the formula –

Surface area  =   pi r l~+~pi r^2 

=   pi r (l~+~r)

 

Let us look at an example to find the surface area of a cone.

Surface area of a cone_2This cone has a slant height of 15 cm and radius of the base is 7 cm. Applying these values in the formula, we get the surface area of the cone

=  pi~*~7~*~(15~+~7)

=  22/7~*~7~*~22

=  484 cm2

 

Surface area of a cone_3Sometimes you are given the height of a cone. In that situation, use Pythagoras theorem to first calculate the slant height of the cone, and then use the slant height to calculate the surface area.

 

 

Example 1 : Calculate the surface area of a cone with height 15 cm and base radius of 8 cm.

Surface area of a cone_4First we need to find the slant height using Pythagoras Theorem.

Slant height l  =  sqrt{15^2~+~8^2}

=  sqrt{225~+~64}

=  17

Surface Area = pi r~(l~+~r)

=  pi~*~8~(17~+~8)

=  3.14 x 8 x 25

=  628 cm2

 

Example 2 : The nose cone of a rocket has a slant height of 5.3 m and a diameter of 6.4 m. Find the area of the curved surface.

Note that you are required to only find the area of the curved surface, not the entire nose cone surface area of the rocket, i.e. you don’t need to find the area of the base.

So, surface area of curved surface  =  pi~r~l

l  =  sqrt{5.3^2~+~3.2^2}  (diameter is given to be 6.4m)

=  sqrt{28.09~+~10.24}

=  sqrt{38.33}

=  6.19 m

Surface Area = pi~r~l

=  3.14 x 3.2 x 6.19

=  62.2 m2

 

Example 3 : Calculate the total surface area of the cone in this figure.

Surface area of a cone_5First we have to find the slant height.

Slant height,  l  =  sqrt{550^2~+~160^2}   (r = 320/2 = 160 mm)

=  sqrt{302,500~+~25,600}

=  sqrt{328,100}

=  572.8 mm

Surface Area = pi~r~l

=  3.14 x 160 x (572.8 + 160)

=  3.14 x 160 x 732.8

=  368,158.8 mm2