How to Calculate the Volume and Surface Area of a Cone

A cone is a 3-dimensional shape with a circular base, and sides that rise up to converge at a point. You can compute its volume and total surface area as long as you know its height and the radius of its base.

Difficulty: Moderate

Instructions

1. 
    

   To find the volume of a cone, use the conical formula for volume:
   
   V = (1/3)πr²h, where r = radius of the base and h = height of the cone.
   
   For example, if a cone has a radius of 5 cm, and height of 12 cm, then the volume is (1/3)π(5²)(12) = (1/3)π(300) = 100π = 314.16 cubic centimeters

2. 
    

   To find the surface area of a cone, we first need to figure out the length of the side of the cone, L. L is related to r and h by the Pythagorean Formula: L² = h² + r².
   
   In our example, we have L² = 12² + 5² = 144 + 25 = 169. Since L² = 169, then L = 13. And so the length of the cone's side is 13 cm.

3. 
    

   Next, consider how a cone is formed. If you cut along the side of a cone in a straight line from the base to the point, and you flatten it out, the shape you get will be one of two things: either a circular sector (pie slice), or a circle with a wedge cut out (pac-man)
   
   A narrow cone will flatten out into a sector. A flat cone will flatten out into a pac-man shape. See image.

4. 
    

   Then, to find the surface area of a cone, you must find the area of the pie slice or pac-man. That area is computed with the formula πLr.
   
   For our example, we have πLR = π(13)(5) = 204.2 square centimeters.

5. Lastly, add the area of the base, which is πr². In our example, π(5)² = 78.54.
   
   So the total surface area of the cone is 204.2 + 78.54 = 282.74 cm².