Hello!
Today we’re going to examine the most common of 3D figures, the rectangular prism, also known as a rectangular solid.
Like with most 3D figures, we can calculate the volume and the surface area by using relatively simple formulas.
But before we do that, we need to define a few terms.
A rectangular prism, or rectangular solid, is a 6-sided object where each side, also called a face, is a rectangle.
It has 12 edges and 8 vertices and all of its angles are right angles.
An important measure of a rectangular prism is the volume.
The volume of a prism or any other 3D object is a measure of how much space it takes up.
We measure this in cubic units, such as cubic inches or cubic centimeters.
It’s easy to picture this with a rectangular prism.
Imagine that we have a bunch of little cubes that are 1 centimeter tall, 1 centimeter wide, and 1 centimeter long.
Each one of these cubes is 1 cubic centimeter, which can also be written like this.
This is our unit of measure.
Notice that it is in cubic units.
Volume is a 3-dimensional measure so it’s always in cubic units.
Now let’s build a rectangular prism out of the little centimeter cubes.
We’ll start with the lowest level and arrange them in a 5 by 3 rectangle.
At this point, we’ve used 15 cubes to make our shape, and we have successfully created a rectangular prism.
But we can also stack the cubes to make it a little taller.
If we place another layer of cubes on top of the first layer, we’ll have a 5 by 3 by 2 rectangular prism.
And since we used 15 more little cubes to make the second layer, we’re up to 30 cubes altogether.
So the volume of our prism is 30 cubic centimeters.
Notice that if we multiply the dimensions of our 5 by 3 by 2 cube, we get our volume!
That’s because the formula for the volume of a rectangular prism is the product of the dimensions:
Volume is equal to, length times width, times height.
So in this case, our volume V is equal to, 5 times 3 times 2, which equals 30.
But let’s not forget our units!
Technically our prism measures 5 centimeters by 3 centimeters by 2 centimeters.
5 times 3 times 2, equals 30, and centimeters times centimeters times centimeters, equals centimeters cubed!
An example of a real-world volume problem is if we needed to pour a foundation for a rectangular building –
we need to measure the area of each side of the building to find out how much concrete we need to prepare.
So if we had to pour a foundation that is 42.5 meters long by 20 meters wide by 0.4 meters deep (this is the height),
we can use those dimensions to find the volume of our foundation by plugging that into our formula:
Volume is equal to, length times width times height.
So our volume is equal to, 42.5 meters times 20 meters times 0.4 meters, which is equal to 340 meters cubed.
Now let’s work on finding the surface area.
Surface area has “area” in the name because it really is a measure of area and therefore is a 2-dimensional measure.
Its units are therefore square units, like centimeters squared.
It’s well-named because we’re literally finding the area of the outer surface of the object.
So all we need to do is find the area of each face, or side, of our rectangular prism and then add all the sides together.
If we take a look at the prism we built from our little centimeter cubes, we can see the squares on the faces.
Each one of these squares is a square centimeter.
We can see 3 of the 6 faces from this angle.
The top face has 15 squares.
The left face has 6, and the right face has 10.
If we add those up, we have 31 square centimeters that we can see.
But since this is a rectangular prism, we know that the sides we can’t see are the same size as the ones we can.
The bottom is the same size as the top.
The back left is the same size as the front right, and the back right is the same size as the front left.
That means that the area of the sides we can't see also totals 31 square centimeters.
Therefore our total surface area is, 31 square centimeters plus 31 square centimeters, for a total of 62 square centimeters.
This problem lets us see the square centimeters, but most surface area problems won’t show us the squares.
We’ll just know the dimensions of the rectangular prism, like this:
Here we can see our prism is 10 meters long by 5 meters wide by 4 meters high.
The corresponding edges on the opposite sides will be the same since this is a rectangular prism.
To find the surface area, I can use the surface area formula for rectangular prisms:
Surface area is equal to, 2 times length times width,
plus 2 times length times height,
plus 2 times width times height.
This formula will give us the surface area if we plug in our length, width, and height.
Don’t be confused about which sides are the length, width, and height.
Depending on how the prism is oriented on the page, it might appear the 10 meters is the length but it really could be the width or the height.
It doesn’t matter!
But for our purposes, let’s say 10 meters is the length, and 5 meters is the width, and 4 meters is the height.
When we substitute all this in, it looks like this:
We have, surface area is equal to, 2 times 10 meters times 5 meters,
plus 2 times 10 meters times 4 meters,
plus 2 times 5 meters times 4 meters.
Notice that we left the units in when we substituted.
Now when we evaluate each term, we get that the surface area is equal to, 100 meters squared plus 80 meters squared plus 40 meters squared.
Notice that our units ended up being square units, which is what we need for the surface area.
Finally, we just add the terms together to get that our surface area is equal to 220 square meters.
Not too difficult, but it’s worth the time to take a moment to look at what that formula is really doing:
The first term, 2 times length times width, is double the length times the width.
This is calculating the area of the bottom and the top of the rectangular prism.
The 2 is there to double the area of either one of the sides.
The middle term, 2 times length times height, is double the length times the height.
In other words, the front and back sides of the prism.
The final term, 2 times width times height, is double the width times the height.
In other words, the left and right sides of the prism.
So the formula is simply finding the area of all 6 sides of the prism 2 at a time.
That means that if we ever forget the formula, we can simply find the area of each side individually and add them all up.
It takes a bit longer but it totally works.
I hope this video on the surface area and volume of rectangular prisms was helpful.
Thanks for watching, and happy studying!