Surface Area Of Cone

 

As shown in the figure below, 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 center at a 90° angle.

when we look at this cone's net, we can understand  about its lateral area and surface areas.

The lateral area of the cone is really a sector of a circle with radius l. The arc length of the sector is the same as the circumference of the base circle.

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.

We are trying to find the area of the sector. The area of the circle with radius l is πl^2. The measure of the arc is the circumference of the smaller circle, 2πr, and the circumference of the bigger circle is 2πl.

Area of sector = πrl

That means the lateral area of a cone is equal to πrl.

To get the total surface area of a cone, let us add the area of the base:

SA = πrl + πr2

DEMONSTRATION:

An  Ice Cream Company is making its own conical waffle cones. The Ice Cream Scooper is a scoop of ice cream that's 6 inches in diameter in a waffle cone. The cone itself has an altitude of 10 inches. How much waffle do they need to make the cone in square inches?

The diameter of the scoop is the diameter of the circular base of the cone. We're interested in the radius, not the diameter. 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
(3)^2 + (10)^2 = 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

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. So we arrive the surface area formula as,

SA = πrl + πr2

SA = (3.14) (3) (98.4) + (3.14)(3)(3) = 926.93+28.26 = 955.19 inches