Video Transcript
The shape in the figure needs to be wrapped in paper. Work out the minimum area of paper that would be needed to completely cover the shape.
To find the minimum area of paper that would be needed to cover the shape, we need to calculate its surface area. That is, we need to find the sum of the area of each of its faces. We can see five faces in this shape, but that’s not all. We have a rectangle underneath the shape, another rectangle at the back of the shape, and another one of these 𝐿 shapes. So we’re going to be finding the sum total of the areas of all eight faces.
Let’s begin by finding the area of the two 𝐿-shape faces. These are called composite or compound shapes because they’re made up of more than one polygon. And so to find their area, we’re going to split them up into these polygons. We could split this shape as shown. Alternatively, we could split it into two different rectangles by dropping this pink line in. We’re going to stick with the first way of separating them though. And so we’re going to find the area of this little rectangle up here and then the big rectangle at the bottom.
The area of a rectangle is found by multiplying its base by its height. This little rectangle has a base of two centimeters and a height of five centimeters. So its area is two times five, which is 10 square centimeters. But what about the area of this second rectangle? It clearly has a width or a base of 12 centimeters, but what is its height? To calculate this, we look at two other parallel lines. We can see that the total height of the shape is 10 centimeters, and the height of the smaller rectangle is five centimeters. This means the height of this other rectangle is 10 minus five, which is five centimeters. And so the area of the rectangle at the bottom is 12 times five, which is 60 square centimeters. The total area of this 𝐿 shape is therefore 10 plus 60, which is 70 square centimeters.
Now remember, we have two shapes identical to one another. So when we add all the areas together, we need to take that into account. We’ll now find the area of rectangle two. We can see it has a width of eight centimeters. But what is its height? Well, in fact, we calculated its height earlier. Its height is the same height as the bottom rectangle in our 𝐿 shape. So that’s five centimeters. And therefore, the total area of this rectangle is eight multiplied by five, which is 40 square centimeters.
We’ll now look at rectangle three. Once again, we can see it has a width of eight centimeters. But what is its other dimension? Well, to calculate this, we look for other parallel lines. Well we see we have this one, which is 12 centimeters, and this one, which is two centimeters. The other dimension in this rectangle is the difference of these. It’s 12 minus two, which is 10 centimeters. The area of rectangle three is therefore eight times 10, which is 80 square centimeters. We move on to rectangle four. This time, we see its height is five centimeters, and once again, its other dimension is eight centimeters. And so the total area of rectangle four is eight times five, which is 40, once again, square centimeters.
We look to rectangle five. Once again, one of its dimensions is eight centimeters, and we saw earlier that its other was two centimeters. Its area is therefore eight times two, which is 16 square centimeters. We’re looking to find the area of two more shapes. That’s six and seven. Shape six is once again a rectangle. It has a width of 12 centimeters and another dimension which is eight centimeters. Its area is therefore 12 times eight. This time, that’s 96 square centimeters.
And we have one more rectangle to consider. That’s rectangle seven, which is at the back of the shape. One of the dimensions of this rectangle is 10 centimeters. We can see its other dimension is eight centimeters once again. The area of rectangle seven is therefore 10 times eight, which is 80 square centimeters.
And so we have the area of all of our shapes. The minimum area of paper that would be needed to completely cover this three-dimensional shape is the sum total of these. We have two 𝐿 shapes. So it’s 70 times two plus 40 plus 80 plus 40 plus 16 plus 96 plus 80, which is 492 or 492 square centimeters.
