If you are struggling with some 3D geometry homework, then this right circular cone calculator will help you find a, the surface area of a cone, and also the following properties:

  • V, the volume of a right circular cone
  • L, the slant height of a right circular cone
  • A_L, the lateral surface area of a right circular cone
  • A_B, the base area of a right circular cone

You only need to know two out of three measurements:

  • base radius
  • cone height
  • slant height

How is a right circular cone defined?

In general terms, a cone is a three-dimensional shape that has a flat base and tapers to a point called the apex or vertex. A right circular cone, in particular, has a circular base with the apex above the center of the base. Typically a right circular cone is what people mean when they say "cone".

If the apex is not above the center of the circular base, it is called an oblique cone. And if the base is a polygon shape (e.g., triangle, square, hexagon, etc.), then it is called a pyramid.

Finding the surface area, a, of a right circular cone

To calculate the total surface area of a cone, we need to add together the surface area of the base and the lateral surface area of the cone. The lateral surface area of a three-dimensional shape is the area that you see from the side. For a cone, this is the area between the base and the apex.

The formula for the surface area of a right circular cone is:

  • A = π * r * √(r² + h²) + π * r²

where:

  • A is the total surface area of a right circular cone

  • π is the famous ratio of the circumference to the diameter of a circle, pi

  • r is the radius of the circular base of the cone

  • h is the height of the apex above the base

Let's go through a worked example for a right circular cone with a base radius of 3 cm and a height of 4 cm:

  • A = 3.1416 * 3 * √(3² + 4²) + π * 3² = 75.4 cm²

You can use any units you like with the calculator, and it will handle the conversions for you. You can also change the units of the results by clicking on the units and selecting from the dropdown menu.

Slant height

If you have a physical right circular cone in front of you that you wish to calculate the attributes of, it might be easier to measure the slant height. The slant height is the distance between the edge of the base and the apex. It's easy to measure with a ruler.

The formula for the slant height is:

  • l = √(r² + h²)

where:

  • l is the slant height of a right circular cone

  • r is the radius of the circular base of the cone

  • h is the height of the apex above the base

This formula derives from Pythagoras' theorem, as you can form a right-angled triangle between the radius of the base and the height of the cone. The slant height is then the hypotenuse of the triangle.

Using the calculator, you can use the slant height instead of the height, and it will calculate rest of the cone's attributes, doing the conversion for you.

For example, for a cone with a base radius of 3 cm and height 4 cm, the slant height is:

  • l = √(3² + 4²) = √(25) = 5 cm

Finding the volume, V, of a right circular cone

The formula for the volume of a right circular cone is:

  • V = (1/3) * π * r² * h

where:

  • V is the volume of the cone
  • π is the famous ratio of the circumference to the diameter of a circle, pi
  • r is the radius of the circular base of the cone
  • h is the height of the apex above the base

This formula also works for oblique cones. For the formula for a truncated cone, check out our cone volume calculator.

Let's show a worked example for a right circular cone of height 4 cm and a base radius of 3 cm:

  • V = (1/3) * 3.1416 * 3² * 4 = 37.7 cm³

Finding the lateral area, A_L, of a right circular cone

The lateral surface area of a three-dimensional shape is the area that can you see from a side-on view. It excludes any flat, horizontal bases or tops. The formula for the total surface area contains the lateral surface area formula, which is:

  • A_L = π * r * √(r² + h²)

where:

  • A_L is the lateral surface area
  • π is the famous ratio of the circumference to the diameter of a circle, pi
  • r is the radius of the circular base of the cone
  • h is the height of the apex above the base

This formula can be simplified, now we know the slant height formula, to:

  • A_L = π * r * l

where:

  • l is the slant height of a right circular cone

So using the result of the example slant height calculation from above (5 cm for a cone of height 4 cm and radius 3 cm), the lateral area of this cone is:

  • A_L = 3.1416 * 3 * 5 = 47.1 cm²

Finding the base area, A_B, of a right circular cone

Since the base of a right circular cone is a circle, then the area for its base is simply the formula for the area of a circle:

  • A_B = π * r²

where:

  • A_B is the area of the base of a right circular cone
  • π is the famous ratio of the circumference to the diameter of a circle, pi
  • r is the radius of the circular base of the cone

For our test cone with a radius of 3 cm, the base area is:

  • A_B = 3.1416 * 3² = 28.3 cm²
Steven Wooding