In these lessons, we will learn

- how to calculate the surface area of a cone when given the slant height.
- how to calculate the surface area of a cone when not given the slant height.
- how to solve word problems about cones.
- how to derive the formula for the surface area of a cone.

**Related Pages**

Surface Area Formula

Surface Area of Prisms

Surface Area of a Sphere

More Geometry Lessons

Geometry Worksheets

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.

The following figures show the formula for surface area of a cone. Scroll down the page if you need more examples and explanations.

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

= πrs + πr^{2}

= πr(r + s)

**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}

**How to find the surface area of a cone when the slant height is given?**

**Example:**

Find the surface area. r = 9cm, h = 12cm, l = 15cm. (Leave answer in π form)

**Example:**

What is the surface area of the cone? Round the answer to the nearest hundredth.

The following video shows how to find the surface area of a cone when not given the slant height.

**Example:**

Find the surface area. r = 7cm, h = 24cm. (Leave answer in π form)

**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.

Example 1: 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?

Example 2: The height of the Pyramid of Menkaure is 216 ft and the side of the (square) base measures 346 ft. Find the lateral surface area.

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

**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.

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

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