Video: Surface Areas of Cones

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

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Video Transcript

In this video, we will learn how to calculate the lateral and total surface areas of cones. We will begin by looking at the definition of a cone and the formulas for its lateral and total surface areas. We will then work through some example questions on surface areas of cones.

Let’s firstly look at the definition of a cone. Cones are three-dimensional geometric shapes that have a circular base and a curved side that ends in a single vertex or apex. A right cone is a cone whose apex lies above the centroid of the base. The centroid is the centre of a circle. The height of a cone is the distance from the apex to the base. This is often known as the perpendicular height.

The slant height of a cone is the distance from the apex to any point on the circumference of the base. This means that the radius perpendicular height and slant height form a right-angled triangle. This information will be useful as we can use Pythagoras’s theorem to help us solve problems involving the surface area of a cone. We will, now, look at the formulas we can use to calculate the lateral surface area and the total surface area of a cone.

The lateral surface area of a cone is the area of a curved surface. This can be calculated using the formula 𝜋𝑟𝑙. This involves multiplying 𝜋 by the radius by the slant height. The total surface area of a cone is the area of all the surfaces, including the base. As a cone only has two faces, the total surface area will be equal to the area of the curved surface plus the area of the base. The area of the curved surface, as already mentioned, is equal to 𝜋𝑟𝑙. As the base of a cone is a circle, its area will be equal to 𝜋𝑟 squared. The total surface area of a cone is, therefore, equal to 𝜋𝑟𝑙 plus 𝜋𝑟 squared. The lateral surface area of a cone is just equal to 𝜋𝑟𝑙.

Let’s look at the example where the radius of the base of the cone is five centimetres, the perpendicular height is 12 centimetres, and the slant height is 13 centimetres. We can calculate the lateral surface area of the cone by multiplying 𝜋 by five by 13. Five multiplied by 13 is equal to 65. Therefore, the lateral surface area is 65𝜋. Multiplying 65 by 𝜋 gives us 204.2035 and so on. Rounding this to one decimal place gives us 204.2. The lateral surface area of the cone is 204.2 square centimetres.

Note that our units are squared and not cubed, even though we have a 3D shape. The units for surface area are square centimetres, square metres, and so on, whereas the units for volume would be cubic centimetres and cubic metres. The total surface area of this cone will be equal to 65𝜋 plus 𝜋 times five squared. We add the area of the curved or lateral surface to the area of the base. Five squared is equal to 25. So we have 65𝜋 plus 25𝜋. This is equal to 90𝜋. Once again, we can type this into the calculator, giving us 282.7433 and so on. Rounding this to one decimal place gives us a total surface area of 282.7 square centimetres. We will now look at some questions involving the lateral and total surface areas of a cone.

Find, in terms of 𝜋, the lateral area of a right cone with base radius nine centimetres and height 13 centimetres.

Let’s begin by drawing a diagram of the cone. We’re told that the base radius is equal to nine centimetres. The height of the cone, which goes from the apex at the top to the centre or centroid of the base, is 13 centimetres. This creates a right-angled triangle with a slant height 𝑙. The lateral area of a cone is the area of its curved surface. This is equal to 𝜋𝑟𝑙. We multiplied 𝜋 by the radius by the slant height. We know that the radius of the cone is nine centimetres. However, we don’t know the slant height at present. We can, however, calculate this by using Pythagoras’s theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse in a right triangle.

In this question, 𝑙 squared is equal to nine squared plus 13 squared. Nine squared is equal to 81. 13 squared is equal to 169. 81 plus 169 is equal to 250. Therefore, 𝑙 squared equals 250. Square-rooting both sides of this equation gives us 𝑙 is equal to root 250. Root 250 is equal to root 25 multiplied by root 10. As root 25 is equal to five, this is equal to five root 10. The slant height of the cone is five root 10 centimetres.

We can now substitute in this value to calculate the lateral area. The lateral area is equal to 𝜋 multiplied by nine multiplied by five root 10. Nine multiplied by five root 10 is 45 root 10. As we’re asked to give our answer in terms of 𝜋, this is equal to 45 root 10𝜋. The lateral area of a right cone with base radius nine centimetres and height 13 centimetres is 45 root 10𝜋 square centimetres. Remember that our units for any area or surface area are square centimetres, square metres, et cetera.

We will now look at another question, calculating the total surface area of a cone.

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

We’re told on the diagram that the height of the cone is 14.5 centimetres. And its slant height is 16.5 centimetres. The radius is currently unknown. We can calculate the length of the radius by using Pythagoras’s theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side of a right triangle, known as the hypotenuse. Substituting in our values gives us 𝑟 squared plus 14.5 squared is equal to 16.5 squared. Subtracting 14.5 squared from both sides gives us 𝑟 squared is equal to 16.5 squared minus 14.5 squared. Square-rooting both sides of this equation gives us 𝑟 is equal to 16.5 squared minus 14.5 squared. This is equal to the square root of 62.

For accuracy, we will leave our answer in this form at present. We were asked to calculate the total surface area of the cone. A cone has two surfaces, a curved surface and a base. Therefore, the total surface area is equal to the area of the curved surface plus the area of the base. The area of the curved or lateral surface is equal to 𝜋𝑟𝑙. We multiply 𝜋 by the radius by the slant height. As the base is a circle, we work out the area of the base by multiplying 𝜋 by the radius squared. Substituting in our values for the radius and slant height gives us 𝜋 multiplied by the square root of 62 multiplied by 16.5 plus 𝜋 multiplied by the square root of 62 squared.

The square root of 62 squared is just equal to 62. As we need to calculate this to two decimal places and not in terms of 𝜋, we can type this calculation into our calculator. This gives us an answer of 602.93801 and so on. The eight in the thousandths column is the deciding number. When this digit is greater than or equal to five, we round up. The total surface area of the cone to two decimal places is 602.94 square centimetres. Any surface area will be measured in square units.

We will now look at the question involving surface area in context.

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

The lampshade is in the shape of a cone with a height of 31 centimetres as shown. The circumference of the base is equal to 145.2 centimetres. We have been asked to calculate the curved surface area of the lampshade. The curved or lateral surface area of a cone is equal to 𝜋𝑟𝑙. We multiply 𝜋 by the radius by the slant height 𝑙. Currently, we don’t know either of these values. We don’t know the slant height, and we don’t know the radius. The circumference of a circle can be calculated using the formula two 𝜋𝑟. We can use this to calculate the radius in this question.

145.2 is equal to two 𝜋𝑟. Dividing both sides of this equation by two 𝜋 gives us 𝑟 is equal to 145.2 divided by two 𝜋. This means that 𝑟 is equal to 23.1092 and so on. For accuracy, we’ll not round this answer at this point. As we know that the radius of the cone is 23.10 and so on centimetres and that the height is 31 centimetres, we can now calculate the slant height. We will do this using Pythagoras’s theorem, which states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side of a right triangle known as the hypotenuse.

Substituting in our values gives us 𝑙 squared is equal to 31 squared plus 23.1092 and so on squared. Typing this into our calculator gives us 𝑙 squared is equal to 1495.0396 and so on. Square-rooting both sides gives us 𝑙 is equal to 38.6657 and so on. We can now substitute the value of the radius and the slant height into the formula for the curved surface area. We multiply 𝜋 by the radius by the slant height. Typing this into the calculator gives us a curved surface area of 2807.132. We need to round this to the nearest square centimetre, which means we need to round to the nearest whole number. The curved surface area of the lampshade is, therefore, equal to 2807 square centimetres.

We will now look at one final example involving surface areas of cones.

A right cone has slant height 35 centimetres and surface area 450𝜋 square centimetres. What is the radius of its base?

We recall here that the surface area of a cone is equal to 𝜋𝑟𝑙 plus 𝜋𝑟 squared. 𝜋𝑟𝑙 is equal to the curved or lateral surface area of a cone. 𝜋𝑟 squared is equal to the base area, as the base of a cone is a circle. Our value for 𝑟 is the radius of the base, and 𝑙 is the slant height. We know that the total surface area is 450𝜋. The slant height is 35 centimetres. Therefore, the curved surface area is 35𝜋𝑟. The area of the base is 𝜋𝑟 squared. As 𝜋 is common to all three terms, we can divide both sides of the equation by 𝜋. This gives us 450 is equal to 35𝑟 plus 𝑟 squared.

Subtracting 450 from both sides of this equation will give us a quadratic equation equal to zero. 𝑟 squared plus 35𝑟 minus 450 is equal to zero. We can solve this by factoring or factorising. We need to find two numbers that have a product of negative 450 and a sum of 35. 45 multiplied by negative 10 is negative 450. And 45 plus negative 10 is equal to 35. This means that our two brackets, or parentheses, will be 𝑟 plus 45 and 𝑟 minus 10.

In order to solve this equation equal to zero, one of the parentheses must be equal to zero. Either 𝑟 plus 45 equals zero or 𝑟 minus 10 equals zero. Solving these two equations gives us 𝑟 equals negative 45 or 𝑟 equals 10. The radius is a length and, therefore, cannot be negative. We can, therefore, conclude that a right cone with slant height 35 centimetres and surface area 450𝜋 square centimetres has a base radius of 10 centimetres.

We will now summarise the key points from this video. A cone is a 3D shape with two surfaces. We have a lateral or curved surface. We can calculate the area of this surface using the formula 𝜋𝑟𝑙. We multiply 𝜋 by the radius by the slant height. A cone also has a circular base. This has an area of 𝜋𝑟 squared. We multiply 𝜋 by the radius squared. The total surface area of any cone is, therefore, equal to 𝜋𝑟𝑙 plus 𝜋𝑟 squared. Surface area is measured in square units such as square centimetres or square metres.

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