Surface Area of Cones
We've covered pyramids (literally, since we found the surface area), and now it's time to cover cones (literally, since we'll find the surface area).
A right cone is a cone where the axis is also the altitude. That means the height from the point on top to the base on the bottom hits the circle dead center at a 90° angle. All other cones are wrong. They're just plain wrong.
If we take a look at this cone's net, we'll be able to say something about its lateral and surface areas other than, "It's right here." Duh.
The lateral area of the cone is really a sector of a circle with radius l. (It used to be the slant height, now it's the radius?) The arc length of the sector is the same as the circumference of the base circle.
Proportions have served us well in the past, and they'll continue to do that if we use them right. The lateral area is the area of the sector. If we compare that to the area of what would be the whole circle, we can compare the arc length to what would have been the circumference.
The area of the sector is what we're trying to find. The area of the circle with radius l is πl2. The measure of the arc is the circumference of the smaller circle, 2πr, and the circumference of the bigger circle is 2πl.
You could go through rearranging and solving yourself, or just trust us that it'll look like this in the end:
Area of sector = πrl
That means the lateral area of a cone is equal to πrl. Unexpectedly simple. To get the total surface area of a cone, we add in the area of the base:
SA = πrl + πr2
The "Bigger Is Better" Ice Cream Company makes its own conical waffle cones. Their Super Duper Ice Cream Scooper is a scoop of ice cream that's 6 inches in diameter in a waffle cone. Mmmm. The cone itself has an altitude of 10 inches. How much waffle do they need to make the cone (in square inches)? And where's the closest store?
The diameter of the scoop is the diameter of the circular base of the cone. We're interested in the radius, not the diameter. (Hopefully he'll have better luck on match.com.) That means our radius r is 3 inches.
What about l, the slant height? The radius and the altitude form two legs of a right triangle with the slant height as the hypotenuse. Pythagorize it up.
a2 + b2 = c2
32 + 102 = c2
109 = c2
c ≈ 10.44 inches
Now that we've found our slant height, we can find the waffle area using the lateral area formula for a cone.
L = πrl
L = π(3 inches)(10.44 inches)
L ≈ 98.4 square inches
Bigger really is better.
Like a pyramid, the surface area of an entire cone (base included), is just the lateral area plus the area of the base.
SA = L + B
We know the lateral area of a cone is πrl. The base of the cone is a circle with area πr2. Plug those in, and we've got a surface area formula.
SA = πrl + πr2
Sweet. Get your spoons poised and your fudge hot and ready. It's ice cream time.