How to Calculate the Volume and Surface Area of a Cone

You can figure out how much ice cream fits inside the cone.

Knowing the volume or surface area of a cone has numerous applications, from architecture to food service (just how much ice cream is inside that waffle cone anyways?). A cone is a three-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.

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, you first need to figure out the length of the side of the cone, L. L is related to r and h by the Pythagorean theorem: L² = h² + r².

   In the example above, L² = 12² + 5² = 144 + 25 = 169. Since L² = 169, then L = 13. The length of the cone's side is 13 cm.

3. 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 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 (like Pac-Man).

4. 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.

   In the example, πLr = π(13)(5) = 204.2 square centimeters.

5. Lastly, add the area of the base, which is πr². In the example, π(5)² = 78.54.

   So the total surface area of the cone is 204.2 + 78.54 = 282.74 cm².

References

• Web Formulas: Surface Area of a Cone
• Web Formulas: Volume of a Cone