# Video: Volumes of Cones

In this video, we will learn how to calculate volumes of cones and solve problems including real-life situations.

14:34

### 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.

Let’s 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, let’s consider its volume. The volume of a cone is equal to one-third 𝜋𝑟 squared multiplied by ℎ. As the base of a cone is circular, 𝜋𝑟 squared refers to the area of the base. ℎ 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.

Let’s 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, we’re 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.

Let’s begin by drawing a diagram of the cone. We’re 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 𝜋𝑟 squared multiplied by ℎ. At this stage, we know the height of the cone, but we don’t 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 we’re 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. We’re asked to calculate the volume of the cone. This is equal to one-third 𝜋𝑟 squared multiplied by the height.

At present, we know the height of the cone, but we don’t know its radius. We recall that the circumference of a circle is equal to two 𝜋𝑟. This means that 318 is equal to two 𝜋𝑟. As we’re told to use 𝜋 as 22 over seven, 318 is equal to two multiplied by 22 over seven multiplied by 𝑟. The right-hand side of our equation simplifies to 44 over seven 𝑟. Dividing both sides by 44 over seven gives us 𝑟 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.

We’re 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 we’re 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 𝜋𝑟 squared multiplied by ℎ, where ℎ 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 𝜋𝑟 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 92𝑟 squared. Next, we can divide both sides by 92 so that 𝑟 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 𝑟 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 𝑟 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 𝜋𝑟 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 𝜋𝑟 squared. When dealing with a square-based pyramid, the volume is equal to one-third multiplied by 𝑙 squared multiplied by ℎ. 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 𝜋𝑟 squared multiplied by ℎ, where ℎ is the perpendicular height from the centre of the base to the apex. As the base of a cone is a circle, 𝜋𝑟 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.