Video: Finding the Volume of a Cylinder given Its Surface Area and Diameter

A cylinder has a curved surface area of 656 cm² and a diameter of 10.2 cm. Find its volume, giving your answer to the nearest cubic centimetre.

03:02

Video Transcript

A cylinder has the curved surface area of 656 centimetres squared and a diameter of 10.2 centimetres. Find its volume giving your answer to the nearest cubic centimetre.

We’re given two pieces of information about this cylinder. We’re told it has a diameter of 10.2 centimetres. We’re also told that its curved surface area is equal to 656 centimetres squared. Our task is to find the volume of the cylinder. So first let’s recall the formula for calculating the volume. It’s 𝜋 𝑟 squared ℎ, where 𝑟 is the radius of the cylinder and ℎ is the height.

We haven’t been given either of these values directly within the question, so we need to use the information we have in order to calculate them. Let’s start with the radius. The radius is always half of the diameter, so if the diameter of the cylinder is 10.2, then the radius is 5.1.

The height is going to be a little bit more complicated to work out, and we need to use the fact that we know — the curved surface area. The curved surface area of a cylinder is found using the formula two 𝜋𝑟ℎ or 𝜋𝑑ℎ. As we know the curved surface area and we know the diameter, we can use these to form an equation that we’ll be able to solve in order to find the height.

So substituting 10.2 for 𝑑 and 656 for the curved surface area, we have 𝜋 multiplied by 10.2 multiplied by ℎ is equal to 656. Now, we could solve this equation completely in order to find the height of the cylinder, but remember we’re then going to substitute it into the formula for the volume. And the volume has a factor of 𝜋 ℎ. So instead of solving for ℎ completely, let’s just so for 𝜋ℎ.

In order to do this, we need to divide both sides of the equation by 10.2. We have that 𝜋ℎ is equal to 656 over 10.2. And we won’t evaluate there any further as we’re now going to use it in our calculation of the volume. So now, we substitute the values of 𝜋ℎ and 𝑟 squared into the formula for the volume. We have the volume is equal to 656 over 10.2 multiplied by 5.1 squared. This gives a value of 1672.8 exactly.

Finally, recall that the question asked us to give the answer to the nearest cubic centimetre. So we need to round this value. Therefore, the volume of this cylinder to the nearest cubic centimetre is 1673 cubic centimetres.

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