### Video Transcript

In this video, we will learn how to
calculate volumes of cones and solve problems including real-life situations. We will begin by giving a
definition of a cone and then explaining the formula that is used to calculate its
volume.

Letβs consider the cone as shown in
the diagram. A cone is a 3D geometric shape that
has 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. When the base is circular, the apex
lies above the circle centre. The height of a cone is the
distance from the apex to the base. This is often called the
perpendicular height. The slant height of a cone is the
distance from the apex to any point lying on the circumference of the base. Finally, the radius, height, and
slant height form a right triangle inside the cone.

Now that we have reminded ourselves
what a cone looks like, letβs consider its volume. The volume of a cone is equal to
one-third ππ squared multiplied by β. As the base of a cone is circular,
ππ squared refers to the area of the base. β refers to the perpendicular or
vertical height from the apex or top of the cone to the centre of the base. The radius is the distance from
this centre to the outside or circumference of the circular base.

Letβs consider an example where the
radius of the circular base is four centimetres and the height of the cone is 12
centimetres. The volume of the cone would be
equal to one-third multiplied by π multiplied by four squared multiplied by 12. Four squared is equal to 16. One-third multiplied by 12 or
one-third of 12 is four. So, weβre left with π multiplied
by 16 multiplied by four. 16 multiplied by four is equal to
64. Therefore, the volume is equal to
64π. We measure volume in cubic
units. Therefore, the volume of this cone
is 64π cubic centimetres. We could work this out as a decimal
answer by multiplying 64 by π. This gives us 201.0619 and so
on. Rounding to two decimal places
gives us a volume of 201.06 cubic centimetres.

We will now look at some specific
questions that involve finding the volume of a cone.

Work out the volume of a cone with
a diameter of 10.5 and a height of 11.3. Give your solution to two decimal
places.

Letβs begin by drawing a diagram of
the cone. Weβre told that the diameter is
10.5. This is a distance from one side of
the circular base to the other, passing through the centre. The height of the cone is equal to
11.3. This is the distance from the apex
or top of the cone to the centre of the base. In order to calculate the volume of
a cone, we use the formula one-third ππ squared multiplied by β. At this stage, we know the height
of the cone, but we donβt know the radius.

The radius of the circular base is
the distance from the centre to the circumference of the circle. This is equal to half of the
diameter. Therefore, to calculate the radius,
we divide 10.5 by two. This is equal to 5.25. And we now have values for the
radius and height of the cone. Substituting in these values gives
us a volume equal to one-third multiplied by π multiplied by 5.25 squared
multiplied by 11.3. Typing this into the calculator
gives us 326.1562 and so on. We were asked to round our solution
to two decimal places. As the six in the thousandths
column is greater than five, we will round up to 326.16. Volume is measured in cubic
units. As there are no specific units in
this question, the volume of the cone is equal to 326.16 cubic units.

We will now look at a second
question, where weβre given the height and base perimeter of a cone.

Determine, to the nearest tenth,
the volume of a right cone having a height of 106 centimetres, given that the
perimeter of its base is 318 centimetres. Use π equal to 22 over seven.

The cone in this question has a
height of 106 centimetres. This is the distance from the apex
to the centre of the circular base. The perimeter of the base is 318
centimetres. As the base is a circle, this is
the circumference of the circle. Weβre asked to calculate the volume
of the cone. This is equal to one-third ππ
squared multiplied by the height.

At present, we know the height of
the cone, but we donβt know its radius. We recall that the circumference of
a circle is equal to two ππ. This means that 318 is equal to two
ππ. As weβre told to use π as 22 over
seven, 318 is equal to two multiplied by 22 over seven multiplied by π. The right-hand side of our equation
simplifies to 44 over seven π. Dividing both sides by 44 over
seven gives us π is equal to 1113 over 22. We can now substitute this value
along with the height into our formula for the volume of a cone. The volume is equal to one-third
multiplied by 22 over seven multiplied by 1113 over 22 squared multiplied by
106. Typing this into the calculator
gives us 284219.727 and so on.

Weβre asked to give our answer to
the nearest tenth. Therefore, the deciding number is
the two in the hundredths column. As this is less than five, we round
down. The volume of a right cone with
height 106 centimetres and base circumference of 318 centimetres is 284219.7 cubic
centimetres. Note that our units here are cubed
as weβre dealing with volume.

Our next question involves
calculating the radius of the cone, given its heights and volume.

A cone has a perpendicular height
of 92 inches and a volume of 420π cubic inches. Work out the radius of the cone,
giving your answer to the nearest inch.

In order to answer this question,
we need to recall the formula for the volume of a cone. This is equal to one-third ππ
squared multiplied by β, where β is the perpendicular height of the cone. In this question, we know that the
volume is 420π cubic inches. The height of the cone is 92
inches. This means that 420π is equal to
one-third ππ squared multiplied by 92. We can divide both sides of this
equation by π. We can also multiply both sides by
three. This gives us 1260 is equal to 92π
squared. Next, we can divide both sides by
92 so that π squared is equal to 1260 over 92. Finally, we square root both sides
of the equation. The radius of the cone is equal to
the square root of 1260 over 92. Typing this into the calculator
gives us 3.70076 and so on.

As we want the answer to the
nearest inch, we need to consider the digit in the tenths column. This is greater than five. So, we round up. And π is equal to four. The radius of a cone with
perpendicular height 92 inches and a volume of 420π cubic inches is four inches to
the nearest inch. We could check this answer by
substituting our value for π back into the initial formula.

The final question we will look at
involves comparing the volumes of a cone and a pyramid.

Which is greater in volume, a right
cone having a base radius of 25 centimetres and a height of 56 centimetres, or a
right-square-based pyramid having a base with a perimeter of 176 centimetres and a
height of 48 centimetres?

Before starting this question, it
is worth recalling that the volume of a cone and the volume of a pyramid have the
same basic formula. The volume of both of these 3D
shapes is equal to one-third of the base area multiplied by the height. When dealing with a cone, this
becomes one-third ππ squared multiplied by height. This is because the base area of a
cone is a circle and the area of a circle is equal to ππ squared. When dealing with a square-based
pyramid, the volume is equal to one-third multiplied by π squared multiplied by
β. This is because the area of a
square is equal to its length squared. The cone has a base radius of 25
centimetres and a height of 56 centimetres. Therefore, its volume is equal to
one-third multiplied by π multiplied by 25 squared multiplied by 56. Typing this into the calculator
gives us 36651.914 and so on. The volume of the cone to one
decimal place is 36651.9 cubic centimetres.

We know that the perimeter of the
square is equal to 176 centimetres. As each side of the square is equal
in length, each length will be 176 divided by four. This is equal to 44
centimetres. As the height of the square-based
pyramid is 48 centimetres, its volume will be equal to one-third multiplied by 44
squared multiplied by 48. This is equal to 30976. Therefore, the volume of the
square-based pyramids is 30976 cubic centimetres. We note that our units for volume
are always cubed. As 36651.9 is greater than 30976,
the shape that has the greater volume is the cone.

We will now recap the key points
from this video. The cone is a 3D shape with a
circular base and a curved side that ends in a single vertex or apex. The volume of a cone is equal to
one-third ππ squared multiplied by β, where β is the perpendicular height from the
centre of the base to the apex. As the base of a cone is a circle,
ππ squared corresponds to the area of the base. Volume is measured in cubic units,
for example, cubic centimetres, cubic metres, or cubic inches. We have used our formula to solve
problems involving the volume of a cone. And we have also compared the
volume of a cone to other 3D shapes.