Lesson video

Hello, everyone, I am Mr. Gratton.

Welcome and thank you for joining me in this lesson on the surface area of 3D shapes.

Today, we will be looking at the surface area of a cylinder and solving problems involving cylinders.

Pause here to have a quick look at some definitions that we'll be using today.

First up, let's look at all of the information that we need in order to find the surface area of a cylinder.

Aisha here asks a good question, what does surface area even mean for a cylinder? Because not every face is a polygon, so we cannot just find the area of each polygonal face and then sum together the areas of each face, right? Well, actually Sam notices that there are three two dimensional shapes on this cylinder.

These shapes are not necessarily polygons, however, because two of these shapes are congruent circular faces that lie on two opposite ends of the cylinder.

These two two dimensional shapes are quite easy to spot.

But what's the third shape? Well, Sam is absolutely spot on.

That curved surface of the cylinder is just a rectangle that has been folded into a cylindrical shape.

Aisha doesn't quite believe what Sam's observed yet, but let's have a look as to why Sam is absolutely right.

Imagine a cylinder without the two circular faces at both ends called a tube.

Imagine a straw or a pipe.

Let's cut from one end to the other and then unfold it.

The result is this flat rectangle.

The 3D curved surface can be unfolded into a two dimensional rectangle.

The height of this rectangle is just the same as the height of the cylinder.

Let's call both of them H.

We can see this because the height of the cylinder is preserved as it is unfolded.

What's harder to notice is this, the length of the rectangle is the circumference of the circular base after unfolding.

Imagine that circular ring at the top of the cylinder is cut and unfolded into a straight line segment.

The length of the perimeter of that circle, its circumference, is the same as the length of the straight line segment it unfolds into.

If the radius of the circle is R, then the circumference of that circle is two pi R.

Therefore, the length of the rectangle is also two pi R.

The area of this rectangle is the product of two pi R and H.

The curved surface area of the cylinder has an area of two pi R multiplied by the height.

Note that this is the area of only the curved surface, this does not include the two circular faces on a cylinder.

Nice for this check, we have a cylindrical tube that has been cut and unfolded into this rectangle.

Using the information from the cylinder, pause here to find the values of A and B.

A is the height of the cylinder at 15 centimetres whilst B is the radius of the cylinder at eight centimetres.

Right, pause here to use these values to find the area of this rectangle.

Two pi multiplied by eight is the width of the rectangle whilst 15 is its height.

The area is therefore the width multiplied by the height at 240 pi, which is 754 centimetres squared.

For this, different cylinder that has a circular face with a circumference of 100 centimetres, pause here to calculate the area of this rectangle.

The circumference is the width of the rectangle at 100 centimetres.

The area of the rectangle is therefore 860 centimetres squared.

The surface area of any 3D shape is the sum of the areas of every single part of that shape.

The curved surface of a cylinder is only one of the parts that we need to consider for any closed cylinder.

As Aisha says, the surface area is the area of the two circles and the area of the rectangle, which is the curved surface, but unfolded.

Then each of these three areas is summed together.

The area of each circle is pi R squared with a radius of 10.

This is true for each of the two congruent circles.

The area of the curved surface is two pi R for the circumference of the circle multiplied by the height of the cylinder.

The area of each circle is 100 pi, and the area of the curved surface is 480 pi.

This means that the total surface area is 680 pi or 2,140 centimetre squared after rounding to three significant figures.

Okay, here's a cylinder that has been broken down into its three surfaces, the two circles and the curved surface.

Pause here to find the area of each of its three different parts.

Each circle is pi R squared with a radius of five centimetres, whilst the area of the rectangle is the circumference of the circle at 10 pi multiplied by the height of the cylinder at 12 centimetres, giving 120 pi.

Right, pause here again to think about what we need to do with each of these values in order to find the surface area of the whole cylinder.

We add each area together to get the total surface area of 170 pi, which is 530 centimetre squared rounded to two significant figures.

Sam has spotted that we can make our calculations a little bit more efficient.

Because the two circles on the cylinder are congruent to each other, we can just find the area of one circle and then double the area to account for the other one.

So, for this cylinder, one circle has an area of pi multiplied by 15 squared.

We can then double that to account for the other circle.

As before, the curved surface can be unfolded into a rectangle, which is the circumference of the circle multiplied by the height of the cylinder.

The area of the two circles when combined is 450 pi, and the curved surface is 1,230 pi, meaning that the surface area of the whole cylinder is 1,680 pi or 5,280 centimetre squared after rounding.

For this check, pause here to find the values of A and B.

The area of one circle is 400 pi, and so the area of two circles is 800 pi.

The circumference of the circle is 40 pi, meaning that the area of the whole curved surface is 840 pi.

Using this information, pause here to find the total surface area of that cylinder.

The sum of these two values is 1,640 pi or 5,152 centimetre squared after some rounding.

Sam goes one step even further beyond.

We can look at the areas of each part of a general cylinder, one with a radius of R and a height of H.

The area of a general circle is pi R squared, which is then doubled because there are two congruent circles on any cylinder, giving us a total of two pi R squared.

Onto the curved surface.

The curved surface has an area that is the circumference of the circular face at two pi R multiplied by the height of the cylinder at H.

Therefore, the surface area of the whole cylinder is the sum of these two values.

This is the formula for the surface area of any cylinder.

Okay, pause here to put this formula into action.

Find the surface area of this cylinder.

This is the formula with a radius and height substituted in.

11 squared is 121, and each part is then evaluated to give a total of 1,232 square yards.

And finally, pause here to identify whose calculation to find the surface area of this cylinder is correct.

Sofia has the only correct answer.

Pause here to think about or discuss what possible mistakes the other three students might have made.

Great stuff.

Onto the practise.

For question one, find the values of X and Y, and then use those values to calculate the area of the rectangle.

And for question two, find the area of the two circles and the rectangle, and then use these values to find the total surface area of the cylinder.

Pause here to do these two questions.

Next up, question three.

Complete this table of information about this cylinder in order to find its surface area.

And for question four, find the surface area of all of these cylinders where one of those cylinders is given as only a description.

Pause here for questions three and four.

Awesome work, good effort, everyone.

The answers to question one, X is 41 and Y is 16.

The area of the rectangle is 1,312 pi centimetres squared.

For question two, each of the two circles has an area of 900 pi, and the rectangle has an area of 420 pi, giving a surface area of 7,000 square inches.

For question three, the combined area of the two circles is 648 pi, and the area of the curved surface is 1,656 pi.

The surface area of the whole cylinder is therefore 2,304 pi or 7,240 millimetres squared.

And finally, question four.

Cylinder A has a volume of 770 centimetres squared.

Cylinder B has a volume of 7,100 centimetre squared whilst cylinder C has a volume of 4,500 centimetres squared.

Now that we've seen how to calculate the surface area of a closed cylinder, let's have a look at some problems where we need to use the surface area of the cylinder to find out other information.

It is possible to map information about a cylinder such as lengths or areas of a cylinder onto two dimensional faces of that cylinder, including the rectangle that is the closed surface of the cylinder.

Sometimes doing this will help find the surface area of that cylinder.

For example, a cylinder has a radius of 14 centimetres and a height of 23 centimetres.

We can label the radii on the two circles, but we can also use this radius to calculate the circumference of the cylinder, which is also the width of this rectangle at 28 pi centimetres.

The height is only used on the rectangle.

The area of each circle is 196 pi, and the area of the curved surface is 644 pi.

We can sum these three areas together to get 1,036 pi centimetre squared as the surface area of the cylinder, even though we have not been given the cylinder itself.

Okay, here's some slightly different information about a cylinder.

Be careful, pay attention to the details given.

Pause here to use all of this information to find the values of A to E.

The radii are 10 centimetres each, half of the diameter of 20 centimetres.

The circumference is 20 pi and the height is 35 centimetres.

This gives a surface area of 900 pi centimetre squared.

This is because the area of the rectangle is 700 pi centimetres squared.

Sometimes the information that we are given is different.

Maybe we need to find the radius or the height of a cylinder if we are given other information about that cylinder.

This is where it is especially helpful to consider each part of the cylinder separately.

For this cylinder, we are given its height and the area of the curved surface only, not the radius, this is currently unknown, so let's call the radii, R.

Both of these values are linked to the curved surface, both the height and the area of that curved surface.

So, let's label what we can on that rectangle, which is the curved surface.

Its height is seven centimetres and its width is two pi R since we currently do not know the radius.

But we can find the radius, let's set up and solve an equation.

This one uses the area of the curved surface as part of the equation.

The area of 56 pi equals the product of the width and the height of this rectangle.

We can then divide both sides by 14 pi, which solves this equation to give a radius of four centimetres.

This radius can then be applied to both of those circles.

Now we can find the area of each of the two circles.

This gives a total surface area of 88 pi centimetres squared.

For this check question, we are given only the height of a cylinder and the circumference of its circular base.

Pause here to find the values of A to D, some of the other properties of that cylinder.

We can use the circumference to find the radius at 15 centimetres.

If the radius of the circle is 15 centimetres, then its area is 15 squared pi or 225 pi.

The area of the curved surface is actually just the height and the circumference multiplied together.

12 multiplied by 30 pi equals 360 pi.

The surface area of the cylinder is the area of the two circles plus the area of the curved surface at 810 pi.

Sometimes you are given the total surface area.

If we're also given the radius, we can use both pieces of information to find the height of the cylinder.

Let's use this information as much as possible across all parts of the cylinder.

The radius is six centimetres, which means the area of each circle is 36 pi centimetres squared.

The circumference of the circle is 12 pi since the radius of six centimetres also applies here as well.

However, since we do not know the height of the rectangle, we need to leave the area of the rectangle as the expression, 12 pi multiplied by H.

The surface area is given as 168 pi, and so 168 pi equals the sum of the areas of the two circles and the area of the curved surface rectangle.

We can subtract two lots of 36 pi from both sides, giving 12 pi H equals 96 pi, and then we can divide through by 12 pi, giving a height of eight.

The height of the cylinder is eight centimetres.

Okay, let's look in detail at all of this information for a cylinder.

Pause here to find expressions for the areas of A and B, and write your answers in terms of either pi or H, the height, or both.

Since the radius is 10 centimetres, the area of each of the circles is 100 pi.

However, since we do not know the height of the curved surface, the area of the rectangle is 20 pi H for some unknown height of H.

However, we can find the value of H, the height of the cylinder, by using the surface area, a 480 pi in that equation.

Pause here once more to fill in the blanks of this equation that involves the surface area and the height of the cylinder, as well as the area of each of the two circles.

The total area of the two circles is 200 pi.

The area of the curved surface is 20 pi multiplied by the height, which we currently do not know, and the surface area is 480 pi.

Now that we have our equation that uses the radius and the surface area of the cylinder, pause here to solve the equation and find the height of the cylinder.

By solving this equation, we have a height of 14 centimetres.

Earlier, we found the formula for the surface area of any general cylinder, so it would be a shame not to use it again for these more problematic questions.

We can substitute known information into this formula and then solve the resultant equation in order to find the height or even the radius of that cylinder.

So, here's the formula again.

We know the surface area is 392 pi and the radius of seven centimetres, which is substituted into two different parts of the formula.

We can evaluate and simplify parts of the equation where possible, remembering the order of operations where we have seven squared as 49 and then multiplied by two for the 98 pi term of this equation.

We can then subtract 98 pi from both sides and then divide through by 14 pi to give a height of this cylinder at 21 centimetres.

Once more, pause here to use the formula for the surface area of a cylinder in order to find the height of the cylinder rounded to one decimal place.

Here's the formula with the surface area and radius substituted in, which is then solved to get a height of 19.

3 centimetres.

Great stuff.

Onto the practise questions.

For question one, pause here to fill in all missing values with a given information in that table, and then use all of the information that you have calculated to find the surface area of that cylinder.

And for question two, find the area of that rectangle from the area of a circle both from the same cylinder, and then use all of this information to find the surface area of the whole cylinder.

For question three, we are given the surface area of the cylinder along with the circumference of one circle.

Find the radius and height of this cylinder.

Pause now for these two questions.

Onto question four, we have two cylinders, each with a total surface area of 2,850 pi centimetres squared.

What is the difference in the heights of these two cylinders? And finally, for question five, you will need to construct a quadratic equation for this question, and then solve that equation in order to find the radius of this cylinder.

Pause here for these last two questions.

Lovely work, everyone.

For the answers to question one, pause here to compare all of your calculations to those on screen for a cylinder whose surface area is 136 pi centimetres squared.

For question two, the radius of one circle is 14 centimetres, meaning that the circumference of the circle is 28 pi.

Therefore, the area of the rectangle, which represents the curved surface of a cylinder, is nine multiplied by 28 pi or 252 pi centimetre squared.

Therefore, the surface area of the whole cylinder is 644 pi centimetre squared.

For question three, by solving an equation involving both the radius and the surface area, the height of the cylinder is 16 centimetres.

For question four, we can solve two equations, each one finding the height of one cylinder.

A has a height of 80 centimetres, whilst B has a height of 17.

5 centimetres.

This means cylinder A is 62.

5 centimetres taller than B.

However, notice that the radius of B is twice the radius of A.

And finally, we can substitute the height of nine and the surface area of 180 pi into our formula, leaving us with an equation with a term in our squared and a term in R.

This is our quadratic equation we need to solve to find the value of R.

We can divide through by pi and rearrange to get this equation R squared plus nine R, take away 90, equals zero.

This factorises, giving us both R equals negative 15 and R equals six.

However, lengths in this case, the length of a radius cannot be negative, so the only valid answer is six.

The radius is six centimetres.

Great effort, everyone, on all of those questions and in a lesson where we have looked at the three parts that make up the surface area of a closed cylinder, the two congruent circular faces and the curved face that can be unfolded into a rectangle.

The surface area is the sum of the two circles and the rectangle, and can be written as this formula, two pi R squared plus two pi RH, where H is the height of the cylinder.

And lastly, the surface area and either the radius or height of the cylinder can be used in order to calculate the other missing length.

Once again, everyone, I appreciate you all and your effort during this lesson.

I have been Mr. Gratton, and you have been absolutely brilliant.

Until our next maths lesson together, take care, everyone, and goodbye.