A cone is a solid with a circular base. It has a curved surface which tapers (i.e. decreases in size) to a vertex at the top. The height of the cone is the perpendicular distance from the base to the vertex.

The net of a solid cone consists of a small circle and a sector of a larger circle. The arc of the sector has the same length as the circumference of the smaller circle.

Surface area of cone = Area of sector + area of circle

Surface area of a cone when given the slant height

Example:

A cone has a circular base of radius 10 cm and a slant height of 30 cm. Calculate the surface area.

Solution:

Area = πr(r + s)

=

= 1,257.14 cm^{2}

The following videos show how to find the Surface Area of a Cone when the slant height is given.

Surface area of cone when not given the slant height

The following video shows how to find the Surface Area of a Cone When Not Given the Slant.

How to calculate the surface area of a cone when the slant height is not given. The Pythagorean Theorem will be used to calculate the slant height using the radius and height of the cone as the right triangle's legs.

Word Problems about cones

Problem: A cone-shaped roof has a diameter of 12 ft. and a height of 8 ft. If roofing material comes in 120 square-foot rolls, how many rolls will be needed to cover this roof?

Find the surface area of a composite figure that consists of two cones and a cylinder.

Derive the formula for the surface area of a cone

This video shows the derivation of the formula for the Surface Area of a Cone.

This video shows how to derive the formula for the surface area of a cone.
In this lesson, we'll take a cone, slice it, squash it, dissect it, and figure out a formula for its surface area.

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