Lesson Video: Surface Areas of Composite Solids | Nagwa Lesson Video: Surface Areas of Composite Solids | Nagwa

Lesson Video: Surface Areas of Composite Solids Mathematics • 7th Grade

In this video, we will learn how to find the surface area of a composite solid using the formulas for lateral or total surface areas of a single solid.

17:40

Video Transcript

In this video, we’ll learn how to find the surface area of a composite solid using the formulas for the lateral or total surface area of a single solid. We can begin by thinking about a composite solid. This is one which is formed from two or more single solids. Here we can see that there is a cone and a cylinder, but neither of these would be composite solids. However, if we place the cone on top of the cylinder, then we would’ve created a composite solid.

So now let’s review some of the key things about finding the surface area of a solid. When we’re discussing surface area, there are two key phrases, either the surface area, or the total surface area, or the lateral surface area. Considering the surface area first, that’s when we find the total area of the surface of a three-dimensional object.

For example, let’s imagine we have a cube of side length 𝑠. Then to find the surface area, we need to work out the area of all the faces. To work out the area of this front face, we would multiply 𝑠 by 𝑠 to give us 𝑠 squared. And as we know that this cube would have six identical faces, then the surface area of the cube is calculated by six 𝑠 squared. So how is this different to lateral surface area?

Well, the lateral surface area is the sum of the areas of all the faces but excluding the base and top. So if we take the cube as an example, we wouldn’t have the top or the base included in the surface area. Therefore, the lateral surface area of the cube would just be the sum of the areas of four faces, which for a cube of side length 𝑠 would then be equal to four 𝑠 squared.

We’ll now look at some questions involving composite solids which include cuboids.

Work out the surface area of the prism.

In order to find the surface area of this three-dimensional shape, we need to calculate the area of each face and then add them together. As this is a prism, then the lengths here will all be three units. Let’s begin by finding the area of one of the easier faces.

The face marked here as number one is a rectangle. So to find the area of a rectangle, we multiply the length by the width. So here we have a three times five, which would give us 15. And as it’s an area, the units would be square units. Next, the face marked two is also a rectangle. So its area will be the length times the width, which is one times three, giving us three square units.

Next, let’s have a look at calculating the area of this green face marked with the number three. If we consider this face in two dimensions, we can see that it would look something like this. If we consider this as a shape made up of two rectangles, then the lengths here would be equal to five minus two, which would give us three. And therefore, the area of this section would be three square units. The area of the lower rectangle would be found by four multiplied by two, which would give us eight square units. And therefore, the total area of this face would be three plus eight square units, which is equal to 11 square units.

As an alternative, we could’ve calculated the area of this face in a different way, noticing that it’s made up of a large rectangle subtract a smaller rectangle. Then the larger rectangle would have an area of 20 square units coming from the five times four. And the smaller rectangle would have an area of three times three, which is nine square units. And therefore, 20 square units subtract nine square units would’ve given us also the area of 11 square units.

So now we’ve found the area of these three faces. We need to work out the area of the faces that are a little bit harder to see. We can consider the face marked in pink here and numbered as four. The length of this face will also be three units, and the width will be four subtract one, which is three units too. And so the area of this face will be three times three, which is nine square units.

Now let’s look at the face that we can’t see any of, and that’s the one on the base of the shape. The area of this face number five would be found by three multiplied by four, which is 12 square units.

It’s interesting to note here that if we look at our pink faces and we look at number two and number four, then when we add the areas of those two, the three and the nine, we’ll get 12 square units. And that’s because if we were looking down directly onto the top of the shape, we’d see the areas of two and four would be equivalent to the area of the base.

So let’s do the same for the other side of the shape. When we calculated the area of shape one in orange, we found that it was 15 square units. And if we look at the shape from the other direction, we would see that the areas of these other faces — let’s call them six and seven — would also be equal to 15 square units.

And now we have one remaining face to find the area of. Can you see where it is? It’s this one at the back, marked with the number eight. But since this is a prism, we’ve already calculated this area. It would be the same as the area of our face number three, which is 11 square units.

When we’re finding the surface area of a prism, it can be helpful to have a strategy to work through it. Sometimes it might be useful to work from side to the neighboring side or other times to look at the opposite sides. Either way, we want to make sure we find the area of every face.

So now to find our surface area, we add together all the separate areas, which is 15 plus three plus 11 plus nine plus 12 plus 15 plus 11. So our answer for the surface area is 76 square units.

If a part of a cube, whose edge length is seven centimeters, is cut to form a cuboid with side lengths of three centimeters, four centimeters, and four centimeters, find the surface area of the remaining part of the cube.

So here we have a cube with part of it — that’s a cuboid — removed from it. And we’re asked to find the surface area of the remaining part of the cube. If this was a volume question, it would be relatively simple as we’d simply subtract the volume of the cuboid from the volume of the cube. However, to find the surface area, we need to find the area of the individual faces and add them together.

We can begin by labeling the side lengths. We’re told that the cube has an edge length of seven centimeters and the cuboid has side lengths of three centimeters, four centimeters, and four centimeters. It doesn’t matter where we put these lengths on the diagram as we’ll still have the same surface area.

So let’s begin by finding the area of one of the faces on our cube which isn’t affected by the cuboid being removed from it. As this is a cube, we know that each two-dimensional face will be a square. So therefore, to find the area of one face, we simply multiply the length by the length. And here we’ll have seven times seven, which is 49 square centimeters. In fact, we will have three in total of these complete faces. We’ll have one on the side, one at the back, and one on the base of the shape. So the area of these three would be three lots of 49 square centimeters, 147 square centimeters.

Next, let’s have a look at the area of this face marked in green. We could notice that this is formed from a square of side seven by seven subtract a rectangle with an area of four by three. If we call this face four, we could then go ahead and calculate that the area of this face would be seven times seven subtract three times four, which would give us the answer of 37 square centimeters.

However, if we then went on to calculate the area of the next face — let’s call this five, and it’s marked in green — we might then notice that the area of four plus the area of face five would actually add together to equal the same as the area of one of the faces of seven by seven.

Let’s demonstrate how. We can see that the width of this rectangle on face five is three centimeters and the length is four centimeters. Therefore, the area of this face would be 12 square centimeters. And we can see that 37 and 12 would add together to give us 49 square centimeters.

We can do the same for the remaining faces of this cube. If we consider the shape from above and look at the faces marked in blue, then the area of these two faces, which we could call six and seven, would also sum to give us 49 square centimeters. And the final two faces are in the direction which we often find the hardest to visualize. These two areas marked in pink would sum to give us 49 square centimeters.

To find the total surface area then, we add together the area of our three squares. And then we have three more lots of 49. In fact, we could also have simply worked out six times 49, as we had six lots of the area of 49 square centimeters. Therefore, the total surface area is 294 square centimeters.

It’s very important to note that this technique of adding the faces, for example, as we did with faces four and five to get the area equal to the opposite face. Only worked here because we had a cuboid, which is a prism with a constant cross section, subtracted from a cube, which is also a prism. So we need to be careful when we’re looking at three-dimensional shapes and work out the best way to find the surface area.

In the next question, we’ll see this composite solid which involves a cylinder and a hemisphere.

This shape is composed of a cylinder and a hemisphere. Find the surface area of the shape, leaving your answer in terms of 𝜋.

To find the surface area of this shape, that means we need to find the areas of each individual face and then add them together. Let’s begin with the cylinder on the base of this solid. To find the area of this upright section of the cylinder, we can use the formula for the lateral surface area of a cylinder, which is equal to two times 𝜋 times the radius times the height of the cylinder. This face can be modeled by a rectangle with a length of two times 𝜋 times the radius and a height ℎ, which is equivalent to the height of the cylinder. Even if we forget this formula, we can recall that two times 𝜋 times the radius is equal to the circumference of the cylinder, which we then multiply by the height to find the area.

Using this formula to find the lateral surface area of our cylinder then, we have two times 𝜋 times the radius of five and the height of 12. We can simplify this to 120𝜋 as we’re asked to leave our answer in terms of 𝜋. The units here would be square units, which will be square centimeters.

The next face to calculate the area of is the face on the base of the shape. As this is formed from a cylinder, then we know that this face will be a circle. And we can use the formula that the area of a circle is equal to 𝜋 times the radius squared. And so for this circle, we’ll have 𝜋 times five squared, which is 25𝜋 square centimeters.

Of course, a cylinder does have a top face as well. But as it’s hidden underneath a hemisphere, then we won’t include it in the surface area, which leaves us with just one more face to calculate the area of. And that’s the surface area of the hemisphere. We can recall that the surface area of a sphere is four 𝜋𝑟 squared. So therefore, half of that to give us the lateral surface area of a hemisphere would be two 𝜋𝑟 squared.

Notice that this is the lateral surface area of a hemisphere. As if we wanted to calculate the total surface area of a hemisphere, we’ll also need to include the area of the circle on its base. Using this formula then, we can notice that the radius is still going to be five centimeters as this hemisphere is sitting exactly on top of the cylinder. So we calculate two times 𝜋 times five squared, being careful to notice that it’s just the radius five which is squared and not the other parts of the formula too. Simplifying this gives us 50𝜋 square centimeters.

Now we’ve found the area of all three faces, we can add them together to find the total surface area. So we have 120𝜋, which was the lateral surface area of the cylinder, plus 25𝜋, which was the area of the circle on the base, and 50𝜋, which was the lateral surface area of the hemisphere. As we’re asked to leave our answer in terms of 𝜋, then our total surface area is 195𝜋 square centimeters.

In the final question, we’ll look at finding the lateral surface area of a composite solid involving cones.

The buoy below is made of two right circular cones on a common base of radius 27 centimeters. If the cost of an erosion-resistant coat is 300 Egyptian pounds per square meter, find to the nearest tenth the cost of painting the buoy.

So here we have a buoy made out of two cones. To find the cost of painting the buoy, that means we’ll need to work out the surface area of it. In this situation, when we’re finding the surface area of these two cones, we don’t need to worry about this circular section at the center because it won’t be painted. We’re therefore interested in the lateral surface area as we’re simply interested in the surface area of the shape excluding the base.

The formula for the lateral surface area of a cone is 𝜋 times the radius times 𝐿, which is the slant height of the cone. We can begin by finding the lateral surface area of the top cone. Here the slant height is 62 centimeters, and we’re not shown the radius on the diagram. But we were told that these two cones have a common base of radius 27 centimeters.

So we calculate 𝜋 times 27 times 62. Using a calculator, we can evaluate this as 5259.026102 and continuing. And because it’s a surface area, our units will be in squared units, which is square centimeters here.

Next, we calculate the lateral surface area of the lower cone. But we may notice that we have a problem. The length given as 71 centimeters is not the slant height of the cone, but instead the perpendicular height. In order to calculate the slant height, we’ll need to use the Pythagorean theorem. We can take a closer look at the triangle formed within the cone. And we know that there will be a right angle as this cone is a right circular cone.

We know that the height of the cone was 71 centimeters. And the radius, the top length, will be 27 centimeters. We can represent the slant height or the hypotenuse of this triangle by 𝐿. The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. Substituting in our values, we’ll have 𝐿 squared equals 27 squared plus 71 squared. So we have 729 plus 5041, which is equivalent to 5770.

To find 𝐿, we take the square root of both sides. So we’ll have 𝐿 is equal to the square root of 5770. We can keep this value in the square-root form as we’ll use it in the lateral surface area calculation. Inputting this value into our lower cone lateral surface area, we’ll calculate 𝜋 times 27 times the square root of 5770, which evaluates as 6443.198979 and so on square centimeters.

To find the total surface area then, we add together the lateral surface areas of our two cones, which gives us a value of 11702.22508 and so on square centimeters. Here we can see that our surface area is in square centimeters. But the cost of the erosion coat is given in a cost per square meters. To change a value given in square centimeters into one given in square meters, we divide by 10000. So our surface area will be 1.1702 square meters.

Now that we know our surface area in square meters, we’re told that the cost is 300 Egyptian pounds per square meter. We can calculate the total cost then by multiplying 300 by 1.1702, which gives a value of 351.066 Egyptian pounds. And as our final answer is to be to the nearest tenth, we’ll have 351.1 Egyptian pounds.

Let’s summarize what we’ve learnt in this video. We learned that a composite solid is one that’s made up of two or more solids. We saw that there’s a difference between the surface area and the lateral surface area as the lateral surface area excludes the area of the base and the top. And finally, here we list some key formulas for the lateral and total surfaces areas of some shapes. Here are the formulas for a cube, a cuboid, and a cone and, finally, the formulas for the lateral and total surface areas of a cylinder and a hemisphere.

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