Recall from Area of a Cone that cone can be broken down into a circular base and the top sloping part. The area is the sum of these two areas.
The base is a simple circle, so we know from Area of a Circle that its area is given by Where r is the radius of the base of the cone.
If we were to cut the cone up one side along the red line and roll it out flat, it would look something like the shaded pie-shaped section below.
This shaded section is actually part of a larger circle that has a radius of s, the slant height of the cone. (To flatten it, the cone was cut along the red lines, the length of this cut is the slant height of the cone.)
The area of the larger circle is therefore the area of a circle radius s, or
The circumference of the larger circle, radius s is
The arc AB originally wrapped around the base of the cone, and so its length is the circumference of the base. Recall that circumference of a circle is given by Where r is the radius of the base of the cone.
The ratio of area x of the shaded sector to the area of the whole circle, is the same as the ratio of the arc AB to circumference of the whole circle. To see why this is so, see Area of a circle sector.Put as an equation Substituting the values from above: Canceling the 2π on the right and solving for x we get
Finally, adding the areas of the base and the top part produces the final formula: