The surface area of a cone

Now it's time for ice cream again. Leon isn't that crazy about ice cream, but he likes the cone. Therefore, he doesn't care about the volume. Leon wants a cone with the largest area. Which cone should he chose?

An ice cream cone is a right circular cone, except you usually draw a cone the other way, like this. A right circular cone has a circular base and a lateral surface that tapers to a point right above the circle's center. The distance from the vertex of the cone straight down to the base surface is the cone's height. The distance from the vertex of the cone to the circle's outer rim is the cone's side. If you cut a cone along its side and then unfold it to make it flat, it becomes a sector.

The cone's side is the circle's radius, and the circumference of the cone's base is the sector's arc. If you can measure the angle of the sector, you can calculate its area, which is the cone's lateral surface area. But if you can't unfold the cone and measure the angle, there's another way. Imagine that you split the sector into several small triangles. In the beginning, the triangle's base is not close to the sector's arc.

But if you split the sector into many narrow triangles, the base of each triangle will eventually follow the sector's circular arc. The total area of these narrow triangles will then get closer and closer to the area of the sector, and therefore to the cone's lateral surface area. The area of each of the triangles is the base times height divided by two. And the heights of those triangles equal one of the cone's side, S. We can re-write this using a common factor like this: the cone's side divided by two times the sum of all the bases of the triangles.

And now if we add up all the bases, we get the sector's arc, which is the circumference of the cone's base, and that's pi times the radius times two. We are almost there. Just simplify the twos, which cancel out, and there you have it - the formula for the cone's lateral surface. The side times pi times the radius - s pi r, spear if you read it. But wait.

How do you find the cone's side if you can't measure it with a ruler? Look at the cone. The base's radius, the height, and the side form a right triangle. This means that if you have two of the values, you can use the Pythagorean theorem to calculate the third one. The cone's lateral surface area is the side times pi times the radius.

The cone's whole surface area equals the lateral surface area plus the base area. The side of the cone can be calculated using the Pythagorean theorem. All this applies only to the right circular cones. It does not work for cones like this one. [silence]