A cone is a three dimensional object with a plane circular base, a slanted curved surface and one vertex. A cone has 3 dimensions as shown in the figure below: the height, h, the slant height, l, and the radius, r.

In a right circular cone, the height and the radius are perpendicular to each other. Hence the three measures, the height, the radius, and the slant height, of a cone make a right triangle. Then according to the Pythagorean theorem, the length of the slant height of a cone is;

When you open a cone up into its corresponding net as shown below, you can see that the base of the cone opens up into a circle with radius ‘r’ and the curved surface of the cone opens up as part of a circle with radius ‘l’ and an arc of length 2πr (equal to the circumference of the base circle).

Now, the total surface area of a cone = area of the base circle + area of the curved surface

Area of the base circle = π r2

Area of the curved surface of the cone = π r l (to understand why this is, please see below the “Why does it work?” section)

So, the total surface area of a cone = π r2 + π r l = π r (r + l)

Hence, to find the surface area of a cone, follow the steps below:

1. First find the radius of the circular base (r) and the slant height of the cone (l).
2. Now find the sum of the radius and the slant height.
3. Then multiply the sum you got in step 2 by the product of radius and the constant π. That’s your answer.

For example: say you need to find the total surface area of a cone with diameter 14 cm and slant height 13 cm.

Let’s try another example: say you need to find the surface area of a cone with radius 6 cm and height 8 cm.

To find just the curved surface area of a cone, use the formula π r l.

For example: say you need to find the curved surface area of a cone with radius 1.4 cm and slant height 5 cm.

Using the above formulas for the surface area (or curved surface area), you can also find any missing value when surface area (or curved surface area) and any one of the measure is given.

For example: say you need to find the height of a cone that has a total surface area of 216 π cm2 and a base radius of 9 cm.

Surface area of the cone = 216 π cm2 radius = 9 cm

π r (r + l) = 216 π

r ( r + l) = 216

9 (9 + l) = 216

9 + l = 216/9 = 24

l = 24 – 9 = 15 cm

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