Surface area of a cone

The formula for the surface area of a cone is:

$pir^2+pirl$

Where

The formula to work out the area of a cone is pie r squared plus pie r length

This formula is easier to remember if you imagine pulling apart a cone to get

Separating a cone to make finding the area isn't easy because the bottom is a circle and the top is a cone

The area of the base is:

Finding the area of the base is easy it is just a flat circle

The area of the cone top is:

To find the area of the top cone its area of base multiplied by slant length over radius of base

The area of the cone top = Area base x ratio slant to radius

The area of the cone top = $\pir^2\timesratiol/r$

The area of the cone top = $\pir^2\timesl/r$

This goes to:

The area of the cone top = $\pirl$

Summary

Area of a cone = Area of base + Area of cone top

= $\pir^2+\pirl$

Interesting example

And then pulling the cone apart the surface areas would look like this:

Base and cone top

That is two full areas of circles.

Or

$\pir^2+\pir^2$

Which becomes: $2\pir^2$

So the surface area of a cone where the slant is the same as the radius is equal to the area of two base circles $2\pir^2$

But this would never be a cone and therefore doesn’t exist under the heading “cones”.

So the way to remember the surface area of a cone is to think:

$\pir^2+\pir^2$

When it isn’t (which it can’t be as a cone).

Then the surface area of a cone must be:

$\pir^2+\pirl$