Surface Area of a Cone

Cones have surface area, which is useful to calculate for practical purposes like painting a cone-shaped object.
The surface area of a cone involves understanding its components: the base and the lateral surface.

The surface area of a cone is given by the formula: \text{Surface Area} = \pi r l + \pi r^2 where:
- r is the radius of the base of the cone.
- l is the slant height of the cone.
- \pi r^2 represents the area of the circular base.
- \pi r l represents the lateral surface area.

The formula for the surface area of a cone can be understood by relating it to the surface area of a pyramid.
Surface area of a pyramid:
\frac{1}{2} \times \text{Perimeter} \times \text{Slant Height} + \text{Area of Base}
In a cone, the perimeter of the base is the circumference of the circle, which is 2 \pi r.
Therefore, the formula becomes:
\frac{1}{2} \times 2 \pi r \times l + \pi r^2
Simplifying, we get:
\pi r l + \pi r^2
This shows the connection between the surface area of a cone and a pyramid.

Slant Height (l): The distance from the vertex of the cone to any point on the edge of the circular base.
Vertical Height (h): The perpendicular distance from the vertex to the center of the base.
It is crucial to distinguish between slant height and vertical height.

Use the Pythagorean theorem to find the slant height l, since the radius, vertical height, and slant height form a right triangle.
a^2 + b^2 = c^2
where a = r = 6, b = h = 8, and c = l (the slant height).
6^2 + 8^2 = l^2
36 + 64 = l^2
100 = l^2
l = \sqrt{100} = 10

Use the Pythagorean theorem to find the slant height when given the vertical height.
Apply the formula \pi r l + \pi r^2 to find the surface area of the cone.