To calculate the surface area of a cone, you need to add the area of the cone’s base to its lateral area. (The lateral area is a fancy name for the area of the surface that connects the base to the peak; in other words, everything but the base.)
A cone with its height and slant height.
A cone with its height and slant height.The following formula gives you the surface area of a cone.
Here’s a little more background on the lateral area:
The lateral area of a cone is one “triangle” that’s been rolled into a cone shape like a snow-cone cup (it’s only kind of a triangle because when flattened out, it’s actually a sector of a circle with a curved bottom side). Its area is
Now for a cone problem:
Here’s the proof diagram.
For the surface area, the only other thing you need is the slant height, l. The slant height is the hypotenuse of the 30-60-90 triangle, so it’s just twice the radius, which makes it
Now plug everything into the cone surface area formula:
