Volume of Prisms and Cylinders
Objective
Rectangular Prism
The rectangular prism is a little odd, mainly because:
- It’s made up of rectangles, but only the opposing sided rectangles are congruent
- Each rectangle shares two sides that are equal to two other rectangles.
- The lateral sides are all the same length.
An example of a rectangular Prism is like so:
8”
4”
3”
We know this is true, because if we look at the corresponding net:
8”
3”
4”
4”
3”
We can see all of these things are true:
The opposing sided rectangles are congruent
Each rectangle shares two sides that are equal to two other rectangles
And finally, the lateral sides are all the same length
Triangular Prism
The rectangular prism is a little odd, mainly because:
- It’s made up of a pair of congruent triangles, and three rectangles
- Each rectangle is the length of one of the legs of the triangles, and one rectangle is the length of the hypotenuse
- All of the rectangles are the same width.
An example of a triangular Prism is like so:
14”
5”
12”
We know this is true, because if we look at the corresponding net:
9”
5”
12”
14”
12”
We can see all of these things are true:
It’s made up of a pair of congruent triangles, and three rectangles
Each rectangle is the length of one of the legs of the triangles, and one rectangle is the length of the hypotenuse
And finally, all of the rectangles are the same width
9”
9”
CYLINDER
The cylinder is pretty standard.
When you think of a cylinder, think of a pipe:
- It has a base of two congruent circles
- One rectangle that wraps around matching the base which is the diameter of the base times pi long
An example of a cylinder is like so:
7”
12”
We know this is true, because if we look at the corresponding net:
We can see all of these things are true:
It’s made up of a pair of congruent circles that makes the base
14”
14”
One Rectangle that wraps around the matching base which is the diameter of the base times pi long
12”
So, let’s start first with a triangular prism
So, let’s imagine just for a second, that we have a triangular prism
Like so:
And we want the surface area of this prism for some reason.
What would that look like?
Well, we want the area of each surface right?
Meaning we want the area of this rectangle:
And this rectangle:
And this rectangle:
Then we’ll want the area of this base:
And the area of this base:
And then we’ll just add all of those areas together to get the entire surface area of the object.
Seems pretty simple right?
But what happens when it’s a little more complicated?
Should we find an easier way?
Well, to start, let’s look at the net of this shape, and maybe we’ll see something that’ll make this easier to find.
Looking at the net
Remember, we’re starting with this triangular prism here:
Now, let’s give it some measurements:
9”
15”
7”
12”
And take a look at its net:
Now, as we can see from the net, the width of the rectangles are going to be the same as the height of the prism
So, the width is:
7”
And we know that each rectangle side is equal to one of the legs of the triangles.
So:
9”
9”
9”
9”
7”
7”
7”
12”
12”
12”
12”
15”
15”
Now, let’s find the area of the rectangles first.
Then we’ll worry about the triangles
So the area of the rectangles is going to be:
However, that’s a really complicated way of looking at it
What if we factored this?
Well, then, if we factored out the 7, it would look like:
But wait, isn’t 9 + 12 + 15 just the perimeter of the triangles?
Which are the bases of the prism?
Well, if that’s the case, then the lateral area could be boiled down to:
And that’s how you find the lateral area of a prism
So the lateral area (area without the base) of a prism is always going to be:
Now the question becomes
How do we find the total surface area?
Well, let’s look back at our net:
7”
9”
9”
9”
9”
12”
12”
12”
12”
15”
15”
7”
7”
7”
Since we know the lateral area is just the sides that aren’t the base:
Then all we need that’s left is the area of the base!
And, since we know the prism has two bases
Then if we factor that into what we have, we can find the surface area!
So, our new equation would be:
Or, to make it less wordy:
So, for this prism, it would be:
AND THAT’S IT!
So that is how you find the surface area of a prism.
You need to first find the lateral area
Then add 2 times the area of the prism’s bases.
So, let’s take a look at some examples:
NOW, LET’S LOOK AT CYLINDRICAL SURFACE AREAS
So, let’s imagine just for a second, that we have a cylinder
Like so:
And we want the surface area of this cylinder for some reason.
What would that look like?
Well, we want the area of each surface right?
Meaning we want the area of this rectangle wrapped around:
Then we’ll want the area of this base:
And the area of this base:
And then we’ll just add all of those areas together to get the entire surface area of the object.
Seems pretty simple right?
But what happens when it’s a little more complicated?
Should we find an easier way?
Well, to start, let’s look at the net of this shape, and maybe we’ll see something that’ll make this easier to find like last time.
LOOKING AT THE NET
Remember, we’re starting with this cylinder here:
Now, let’s give it some measurements:
9”
3”
And take a look at its net:
Now, as we can see from the net, the width of the rectangle is going to be the height of the cylinder.
So, the width is:
And the length of the rectangle must be the circumference of the base
So:
Now, let’s find the area of the rectangles first.
Then we’ll worry about the circles
So the area of the rectangles is going to be:
But, if we look at what we’re actually doing here
We’re multiplying the height of the cylinder
By the circumference of the base
So, this means that we could boil down our lateral area down to:
9”
9”
(Which is very similar to our last formula, since the circumference of a circle is a fancy way of saying its perimeter).
AND THAT’S HOW YOU FIND THE LATERAL AREA OF A CYLINDER
So the lateral area (area without the base) of a cylinder is always going to be:
Now the question becomes
How do we find the total surface area?
Well, let’s look back at our net:
Since we know the lateral area is just the sides that aren’t the base
Then all we need that’s left is the area of the base!
And, since we know the cylinder has two bases
Then if we factor that into what we have, we can find the surface area!
So, our new equation would be:
Or, to make it less wordy:
So, for this cylinder, it would be:
9”
9”
3”
3”
And that’s it!
So that is how you find the surface area of a cylinder.
You need to first find the lateral area
Then add 2 times the area of the cylinders bases.
So, let’s take a look at some examples:
So what is volume?
So according to Google, the definition of volume is:
"the amount of space that a substance or object occupies, or that is enclosed within a container, especially when great.”
Which is basically true
Volume is how much space something can either hold
Or occupy
Like a gallon of milk
That’s a measurement of volume
Or a cup of sugar
Again, volume
And everything that is tangible (something you can pick up and hold)
Has some volume.
So, let’s take a look at how we can apply that to our three dimensional shapes.
Investigating……
Let’s say you have some prism:
And you want to figure out what volume this prism is
Because you want to carry this pile of cash:
And you want to be inconspicuous
So no one tries to steal it
(Because who’s going to leave that much money in a box?)
How can you tell if the money is going to fit?
Well, you could try to force it
But that takes time
And with all of that cash, you want to make sure it’s secure as quick as possible.
So, how about we take the volume of the container?
Check out how much room the box can hold
Then see if the cash can even fit
Or if you need a bigger box?
(Since I’m part of this “we”, I’m charging a 5% fee)
Finding the volume
So, to start off with, we need some dimensions
So:
10”
7”
5”
Now we want to figure out how many much room this box can hold
So, the way we find the volume is we figure out the area of the side:
And multiply this by the length of the box
Because when we do:
We then take up the entire inside of the box.
So, the area of the side is:
7 * 5 = 35
And multiplying35 by the length of the box:
35 * 10 = 350
So the box can hold 350 cubic inches!
And wouldn’t you know
The stack of cash is only 330 cubic inches
So you’re all good!
AND THAT’S HOW WE FIND THE VOLUME OF A PRISM
To find the volume of a prism, we just multiply:
Volume = Length * width * height
Now let’s talk about cylinders:
FINDING THE VOLUME OF A CYLINDER
Now, the thing about a cylinder is
It’s basically a prism
The only difference between the two is the base
So, looking at a cylinder like this:
We know that to find the volume
We need to find the area of the base:
And multiply it by the length of the cylinder:
So, to find the volume of a cylinder
We can see that the formula would be:
And that’s all there is to it!
However, it’s not always that easy
Mainly because sometimes we have cylinders like this:
And that makes finding the height kinda hard
But, that’s where Cavalieri’s Principle comes in:
CAVALIERI'S PRINCIPLE
So Cavalieri figured out that volume isn’t shape specific
But instead, if we have a change in shape, it could have the same volume
He figured this out using coins
He realized that if you have coins stacked up nice and neat
And you have coins stacked not so neat
They still have the same volume
Which helped because when you have a shape like this:
You can still find the volume as long as you measure the height of the object
And the height of the object is:
So now that we know this
We can solve some examples:
EXAMPLE 1:
Find the volume of the following cylinder:
4”
14”
So, to find the volume of this cylinder, we need to find the area of the base:
Next, we’re going to multiply this by the height of the cylinder:
So the volume of this cylinder is about: 703 cubic inches
Example 2:
Find the volume of the following prism:
So, to find the volume of this triangular prism,
we need to find the area of the triangle:
Next, we’re going to multiply this by the thickness of the prism:
So the volume of this prism is about: 270 cubic inches
14”
5”
9”
12”
Example 3:
Find the volume of the following prism:
So, to find the volume of this rectangular prism,
we need to find the area of the base:
Next, we’re going to multiply this by the thickness of the prism:
So the volume of this prism is about: 96 cubic inches
8”
4”
3”