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
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