### 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.