In this explainer, we will learn how to calculate the lateral and total surface areas of cones using their formulas.

We will begin by reviewing some of the language associated with cones and their areas. There are three key measurements that describe a cone:

- the radius, , of the cone’s circular base, also referred to as its base radius;
- the perpendicular height, , that is the perpendicular distance between the center of the base and the vertex of the cone;
- the slant height, , that is the distance from the vertex to a point on the circumference of the base, along the side of the cone.

These three lengths are represented in the figure below. We will see how these three lengths are related to one another later.

The surface area of a cone is composed of two different parts: the area of the curved surface, which is called the lateral area or lateral surface area, and the area of the circular base.

### Formulas: Surface Areas of a Cone

The formula for the lateral surface area of a cone, , is where represents the base radius of the cone and represents the slant height.

The formula for the total surface area of a cone, , is

We must be careful to make the distinction between these two surface areas in order to determine whether the area of the base is to be included in any specific problem.

We will now demonstrate how to apply the formula for calculating the lateral surface area of a cone given its base diameter and slant height.

### Example 1: Finding the Lateral Surface Area of a Cone given Its Base Diameter and Slant Height

Find, to the nearest tenth, the lateral area of a cone with a diameter of 40 centimetres and a slant height of 29 centimetres.

### Answer

First, we note that the question asks only for the lateral area of the cone, not its total surface area. The formula required is and so we need to know the slant height of the cone and its radius. The slant height of 29 cm is given in the question, and we can calculate the radius by halving the diameter:

Finally, we substitute these values into the formula for the lateral surface area and evaluate:

Rounding our answer to the nearest tenth as required by the question, the lateral
area of the cone is
1 822.1 cm^{2}.

A key point from the previous example was that we were given the diameter, not the radius, of the base of the cone. This presented no significant challenge because the two are easily related, but it is another detail to look out for when approaching a problem.

It was also important that we identified whether we were calculating the lateral or total surface area of the cone. As we were calculating the lateral area, we did not need to include the area of the circular base. Let us now consider an example of applying the formula to calculate the total surface area, given both the slant height and the base radius of a cone.

### Example 2: Finding the Total Surface Area of a Cone given Its Slant Height and Base Radius

Determine, to the nearest hundredth, the total surface area of the cone shown.

### Answer

From the figure, we identify that we have been given the base radius of the cone, which is 19 cm, and its slant height, which is 40 cm. To calculate the total surface area, we substitute both of these values into the relevant formula and simplify:

In some instances, we may be required to give an exact answer, in which case we would leave our answer in terms of . In this problem, however, we are asked to give our answer to the nearest hundredth, so we evaluate further:

The total surface area of the cone, to the nearest hundredth, is
3 521.73 cm^{2}.

We have now seen examples of how to calculate both the lateral and the total surface areas of a cone given its base radius (or diameter) and its slant height. However, we may be required to calculate the surface area of a cone when instead we are given its perpendicular height. We do not need a separate formula, but instead we consider the relationship between the base radius, the perpendicular height, and the slant height.

From the figure above, we see that the base radius, the perpendicular height, and the slant height form a right triangle. The relationship between these three lengths can, therefore, be described by applying the Pythagorean theorem:

Hence, if we know two of these lengths, we can calculate the third by forming and solving an equation. Let us consider an example of this.

### Example 3: Finding the Total Surface Area of a Cone given Its Slant Height and Perpendicular Height

Find the total surface area of the right cone approximated to two decimal places.

### Answer

From the figure, we see that the two lengths given are the slant height and the perpendicular height of the cone. The formula for the total surface area of a cone is and so we need to determine its base radius .

We can form an equation relating the base radius, the perpendicular height, and the slant height by applying the Pythagorean theorem:

Substituting and and simplifying gives

We solve for by square rooting, taking only the positive value as is a length:

Finally, we substitute the radius and the slant height of the cone into the formula for the total surface area. It is preferable to use the exact value for , and indeed , in order to prevent the introduction of any rounding errors:

The total surface area of the cone, to two decimal places, is
602.94 cm^{2}.

We have now seen examples of how to calculate the lateral and total surface areas of a cone given two of its three key measurements. We can summarize these processes in the following steps.

### How To: Calculating the Surface Area of a Cone

- Determine whether the lateral area or total surface area is required.
- Identify the base radius and slant height of the cone.
- If one of these lengths is not given but the perpendicular height is, calculate the missing length by applying the Pythagorean theorem.
- Substitute the lengths of the radius and slant height into the relevant formula and evaluate.

As with all areas of mathematics, the skills we have encountered here can also be applied to problems in a real-world context. Whenever a real-world object can reasonably be modeled by a cone, we can apply the formulas we have introduced to calculate its lateral or total surface area, as we will see in the next two examples.

### Example 4: Finding the Lateral Surface Area of a Cone in a Real-World Context

A conical mountain has a radius of 1.5 km and a perpendicular height of 0.5 km. Determine the lateral area of the mountain to one decimal place.

### Answer

The question tells us that this mountain is conical, and so the problem is essentially a geometric one. In order to apply the formula for the lateral area, we need to know both the base radius and the slant height of the mountain. We are not given the slant height, but as we do know both the base radius and the perpendicular height, we can apply the Pythagorean theorem:

Substituting and and simplifying gives

To solve for , we find the square root, taking only the positive value as is a length:

We now substitute the values for the radius and the slant height of the cone into the formula for the lateral area:

The lateral area of the mountain, to one decimal place, is
7.5 km^{2}.

We should always be careful to ensure we include the correct units for our answer. As we are calculating areas, the units for our answer should be square units. In the previous example, the lengths given were measured in kilometres and so the units for our answer were square kilometres. We should also ensure that we check whether the question has asked for the answer to be given in units that differ from those originally given, for example, if the length units were metres and the area is required in square kilometres.

### Example 5: Finding the Lateral Surface Area of a Cone in a Real-World Context

A conical lampshade is 31 cm high and has a base of circumference 145.2 cm. Find the curved surface area of the outside of the lampshade. Give your answer to the nearest square centimetre.

### Answer

The curved surface area of the lampshade is its lateral area, which is calculated using the formula

We therefore need to know both the base radius of the cone and its slant height, neither of which are given in the question. Let us consider instead the other information we are given and determine how we can use it to calculate the lengths we need.

The circular base of the cone has a circumference of 145.2 cm. We know the formula for the circumference of a circle is and so we can calculate the radius by forming and solving an equation:

We now know both the base radius and the perpendicular height of the cone. To calculate the slant height, we apply the Pythagorean theorem:

We solve for by square rooting:

Finally, we substitute the values we have calculated for the radius and slant height into the formula for the lateral surface area:

The curved surface area of the outside of the lampshade, to the nearest
square centimetre, is
2 807 cm^{2}.

In other problems, we may be given the surface area of a cone and one other piece of information and asked to determine one of the other key lengths. This is essentially “working backward,” as we will see in our final example.

### Example 6: Finding the Height of a Cone given Its Surface Area and the Radius of Its Base

The surface area of a cone is square inches, and the radius of the base is 13 inches. Determine the slant height of the cone.

### Answer

In this problem, we are given the radius and the surface area of a cone and asked to determine its slant height. We recall the formula for the (total) surface area of a cone:

By substituting the known surface area and the known radius, we are able to form an equation:

We can now solve this equation to determine the value of . A factor of can first be canceled from each term to give

We then subtract 169 from each side and divide by 13:

The slant height of the cone is 15 inches.

Let us finish by recapping some key points.

### Key Points

- The three key lengths that describe a cone are its base radius , its perpendicular height , and its slant height .
- These three lengths are related by the Pythagorean theorem:
- The curved surface area of a cone is called its lateral area and is calculated using the formula
- To find the total surface area of a cone, we also need to include the area of the circular base: