

Surface area of a cone  derivation
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
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.
The top
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 pieshaped 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:
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