Surface area of a cone


Here you'll learn how to calculate the surface area of a cone.


As for volume of a cone, let's keep it simple and consider a right circular cone - one which has its apex directly above the center of its circular base.



What is the surface area of a cone?

The sum of the cone's circular base and curved surface areas.


Surface area of cone formula

Cone surface area  =  circular base area  +  curved surface area



If l = slant height and given a cone's base is circular and the area of a circle is πr² (where π= pi and r = radius of circle), the formula becomes



where SAcone = surface area of cone.


Example calculation

What's the surface area of a cone with base radius (r) 3cm and slant height (l) 5cm (where cm = centimeters)?


SAcone

=   πr²  +   πrl

=   x 3²)  +  (π x 3 x 5)

=   75.4cm² (square centimeters) to 1 d.p. (one decimal place)


When working out the above, use the left and right bracket buttons on your calculator.




The key stroke sequence is left bracket, pi sign (π), multiply sign (x), 3, squared sign (x²), right bracket, multiply sign (x), left bracket, pi sign (π), multiply sign (x), 5, right bracket, equals sign (=). Your calculator should display 75.39822369 which is 75.4 to 1 d.p. You may need to access x² and π via a shift or 2nd function button.


What if I only know the circular base radius (r) and perpendicular height (h)?


For instance, r = 3cm and h= 4cm.


Remember we're considering a right circular cone - the radius (r), perpendicular height (h) and slant height (l) form the sides of a right angle triangle, so the Pythagorean theorem can be applied.


l²  =  r² + h²

l  =  √ (r² + h²)  =  √ (3² + 4²)  =  √ (9 + 16)  =  √25 = 5cm


You can now enter r = 3cm and l = 5cm into the surface area of cone formula SAcone = πr² + πrl.


Since the figures are the same as for the prior example calculation, you should get the answer 75.4cm² (square centimeters) to 1 d.p. (one decimal place).


Let's move on!


Click below for more cone geometry:

Volume of a cone


Related topics:

Volume of a cylinder

Surface area of a cylinder


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