Cone Surface Area Calculator

This cone surface area calculator allows you to calculate the lateral surface area, base surface area and total surface area of a right circular cone.

You just have to enter two out of three values (radius, height or slant height) in order to find the missing value and then calculate all areas. The volume is also shown, and if needed, the apex angle and angles at the base can be determined as well.

What is the surface area of a cone?

A cone has two faces; the first is a curved face that runs from a circular rim to an apex at the top. This point is called the vertex. The surface area of this curved face is the lateral surface area and sometimes referred to as the curved surface area or CSA.

The second surface is a flat circle on the ground, so it's the base. If you add up the base and the lateral area, then you get the total surface area that would be required to cover the entire closed cone.

Since each face is two-dimensional, the cone itself is a three-dimensional object, but its faces are measured in square units such as square centimeters or square feet.

Here's how to use this calculator.

Enter any two values of Radius, Height or Slant Height into the corresponding fields and select the desired units. Once you have entered two values, the tool will calculate the third value and display the Lateral Area, Base Area, Total Surface Area, and Volume.

It is not necessary to specify which combination of values you entered. If you know the radius and height of an ice cream cone, enter those two values. If you are measuring a party hat and only have the radius and slant height available, enter those two values. To start over just leave one field blank and enter new values. By checking "Angle" you can also display the half-angle at the tip, base angle, and total angle of the cone.

Formula for surface area of a cone:

It all starts with the slant height. The slant height is a straight line from the tip to an edge of the base. It forms the hypotenuse of a right triangle, where the other two sides are the radius and the height. These can be related using the Pythagorean theorem.

The surface area is equal to the product of pi, the radius and the slant height. Note that the slant height is used instead of the vertical height, which often causes confusion.

Since the base is a circle, its area is equal to pi times the square of the radius.

The total surface area is the sum of the lateral surface area and the base area. By factoring out the common factor of π and the radius, a more compact formula results.

where r is the radius, h is the height and l is the slant height, with π being approximately 3.14159. As a bonus feature, the tool also calculates volume. For the calculation of volume, vertical height is used instead of slant height.

An important difference between mantle line and height:

The most common mistake with cones is using the vertical height instead of the slant height when needed. The vertical height runs straight up along the central axis of the cone, while the slant height runs along the outside surface. The slant height will always be longer than the two.

When calculating the surface area, we focus on covering the outside, so the slant height is used for the lateral area. When calculating volume, we focus on filling the inside, so the vertical height is used. If only the height is known, then the tool first calculates the slant height and then the area, so there can be no confusion between the two values.

Why the curved surface is part of a circle.

If you cut along the slant height of a cone and flatten out the side, you will get a sector of a circle. The radius of this sector is equal to the slant height, and the length of the arc is exactly the circumference of the base of the cone. This is why the surface area of the side is pi times the radius times the slant height. This intuitive picture also helps when you cut out a flat piece of material and then form it into an actual cone.

Example calculation:

Suppose we have a cone with base radius of 8 centimeters and height of 15 centimeters. First, let's calculate the slant height using the Pythagorean theorem.

Then we calculate the surface area.

The area of a circle is equal to pi times the radius squared.

If you add both of them together, you get the total area.

Cone formulas summary

The following table summarizes the individual relationships used in a calculation tool. A cone is determined by two of three linear measurements. So if two of these values are entered, they will be sufficient to determine the remaining value.

Surfaces of different cones

All of these cones have a radius of 5 centimeters. You can see that the lateral surface area increases as the height of the cone increases while the base area stays the same. The values are rounded to two decimal places.

Applications of surface area of a cone

If you look closely, you'll notice that cones are everywhere. Ice cream waffles, traffic cones, party hats and funnels are all cones, as are the conical roofs of gun turrets or the top covers of silos.

If the lateral surface area is known, then manufacturers can determine how much metal sheeting, paper, or fabric they need to cover a cone. This is a fundamental calculation in making paper drinking cups, speakers, lampshades, and rice straw hats in Southeast Asia. For closed cones, total surface area is important because it indicates the amount of material needed while volume indicates how much an object can hold.

Tips for accurate results:

Measure the radius to the center or measure the width across the base and divide by two. Check before entering whether your measurement is the vertical height or slanting side length, as these are different lengths. Be aware that units must be consistent and area increases in proportion to the square of dimensions so small errors in length can lead to large errors in area.