This surface area of a cone calculator will help you calculate the surface area of any right cone. In the text below, we will show you the surface area of a cone formula and how to derive it. After using our calculator and reading this article, you'll be super confident about how to find the surface area of a cone.

What type of cone is this calculator useful for?

Before we explain how to use our surface area of a cone calculator, let's first define what type of cone you can use it on. A generic cone shape consists of a circular or oval base and a vertex (or tip) above the base connected to the perimeter of the base shape.

This calculator is for a particular type of cone called a right cone (as shown in the diagram above the calculator). It has a circular base, and the vertex is directly above the center of the base. Therefore, the angle made between the base and an imaginary line between the base and the vertex is 90°, commonly known as a right angle.

This tool cannot be used for oblique cones, where the vertex is off-center. For those that are comfortable with advanced calculus, you can learn more about calculating the surface area of an oblique cone by reading this discussion about the problem.

How to use this surface area of a cone calculator

The calculator is quite simple to use. Let's go through it step-by-step:

  1. Enter a value for the radius of the circular base. Remember that the radius is half of the diameter of a circle. You can choose different units of length, depending on the problem or measurement taken. Alternatively, you can enter the circumference of the circular base instead.

  2. Enter the height of the cone or the slant height of the cone, depending on which one is known. The height is the perpendicular distance between the cone tip and the center of the circular base. The slant height is the distance between the tip and the outside edge (perimeter) of the base.

    Keep in mind that the height of the cone needs to be greater than zero and that the slant height (if you are entering it) needs to be longer than the radius of the base.

  3. The surface area of the right cone will then be displayed. You can change the units of the area, depending on your taste or the overall size of the cone.

How to derive the surface area of a cone formula

To gain a greater understanding of how this calculator works, we will now have a look at what the formula of the surface area of a cone is and how to derive it.

The first step in tackling the problem is to split the right cone into two different parts; the circular base and the side area of the cone (the lateral area). Let's start by calculating the area of the circular base.

The area of a circle is given by:

  • Abase = πr²

where:

  • Abase is the surface area of the base of the cone
  • π is the ratio of the circumference to the diameter of a circle
  • r is the radius of a circle

Later, we will add this base area equation to the equation we will derive next for the lateral area of a right cone.

A cone folded flat forms a sector of a larger circle.

Imagine a cone without its base, made out of paper. You then roll it out so it lies flat on a table. You will get a shape like the one in the diagram above. It is a part (or a sector) of a larger circle, whose radius (l) is equal to the slant height of the cone. The arc length of the sector (c) is equivalent to the circumference of the cone base.

By combining the equation for the area of a sector in terms of radius and angle with the equation for the arc-length of a sector, we can write the area of the sector as:

  • Alat = (1/2)cl

where:

  • Alat is the lateral area of the cone
  • c is the arc-length of the sector and therefore the circumference of the cone
  • l is the radius of the sector and the slant height of the cone

Now by using the circumference of a circle formula c = 2πr, the above equation can be rewritten in terms of the radius of the cone's base:

  • Alat = πrl

Adding the lateral area to the result for the area of the base of the cone, we get A, the total surface area of a right cone:

  • A = (πr²) + (πrl)

This can then be simplified to:

  • A = πr(r + l)

If you only know the perpendicular height of the cone, you can use the equation for the slant height in terms of the radius to get:

  • A = πr(r + √(h² + r²))

where:

  • h is the perpendicular height of the cone

Examples of how to find the surface area of a cone

We now have two equations to calculate the surface area of a right cone, depending on what parameters are known. Let's look at some worked examples.

For a cone with a base radius of 3 inches and a height of 4 inches, we can calculate the surface area like so:

  • A = πr(r + √(h² + r²))
  • A = π * 3 * (3 + √(4² + 3²)) = 75.4 in²

Let's take another cone where we measure the diameter of the base to be 10 inches, and the slant height is 15 inches. First, divide the diameter by two to get the radius, 5 inches. Then use the formula for the area of a cone with slant height, like this:

  • A = πr(r + l)
  • A = π * 5 * (5 + 15) = 314.16 in²

Now you know how to find the surface area of a cone and the equations that power this surface area of a cone calculator.

If you need to calculate the surface areas of other 3D geometric shapes, why not try out our surface area calculator.

Steven Wooding