Cross Sections and Solids of Rotations
Objective
The move from 2-D to 3-D objects
So far, we’ve dealt with all sorts of two dimensional objects
From triangles
To quadrilaterals
To circles
To triangles in circles
To quadrilaterals in circles
Now that we’ve pretty much explored as much as we can with two dimensional objects
And we still have 8 weeks of school left
Let’s start working with 3 dimensional objects
So, in order to start, we need to go over a few deginitions:
Definitions
A net – a mathematical net is a diagram of the surface of a three-dimensional figure that can be folded to form the three-dimensional figure.
Basically think of it as an outline, something like:
See, the reason this is considered a net is because if we fold this along the lines provided
Then we can see we actually create a three dimensional shape, like:
The base of a shape is the top and bottom of the shape, that are the only two sides that are the exact same.
So, for example, these are our bases:
The lateral sides of a shape are the rest of the sides of the shape.
Basically, think of them as the sides without the bases.
So, for example, these are our lateral sides:
Now that we’ve talked about the different parts of a three dimensional shape, let’s talk about the different kinds of shapes we may see:
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”
RECTANGULAR PYRAMID
The rectangular pyramid is actually what you think of when you think of a pyramid.
The rectangular pyramid has:
- A base made up of a rectangle
- Two congruent opposing isosceles triangles
- All triangles have the same length as their corresponding rectangle’s side
An example of a rectangular pyramid is like so:
11”
6”
We know this is true, because if we look at the corresponding net:
6”
11”
We can see all of these things are true:
It has a base made up of a rectangle
Two congruent opposing isosceles triangles
And finally, all of the triangles have the same length as their corresponding rectangle’s sides
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”
CROSS SECTION
So now that we’ve covered our three dimensional shapes, let’s talk about cross sections.
A geometric cross section is a region of a plane that intersects a solid figure.
That’s a fancy way of saying it’s like cutting the three dimensional shape, then looking at the part you cut.
An example is like:
As we can see, it looks like the cube is being cut by the paper,
And what we’re looking at is the left over of the cut.
However, these can get weird looking too.
Such as:
Now we can see the cube cross section look more like a rectangle.
Which is weird, but not
It’s all about the angle you put the plane (paper) and the object you are sectioning off.
Like this:
SO WHY DO WE CARE?
Well, basically because there are many times in life when pieces just don’t fit together.
Life isn’t a giant perfect puzzle where you just have to find the right piece
Sometimes, you have to make the right piece
This is used in
3D printing
Carpentry
Engineering
Architecture
Basic home maintenance
So, here’s an example of what to expect:
EXAMPLE 1:
Describe the cross section of the following:
F
D
C
B
E
H
I
J
As we can see here, it looks like we have a cube
And the cross section is at an angle
However, when looking at the cross section
It looks remarkably like a square
So, our cross section here is a square.
SOLIDS OF ROTATIONS
Now let’s talk about solids of rotations
So, we already know what a rotation is from last semester
But that’s only thinking in 2 dimensions!
We need to think in 3 dimensions now
So, imagine you have a two dimensional object:
And you take it, and twirl it about it’s y axis:
Well, then you’d get something like:
And that is a solid rotation
SO, THAT’S IT?
That’s it!
The only thing is, this is used A TON
It’s used to make cars, when things spin to get your tires to move
It’s used in airplanes
In computers
Basically anything that spins
The reason we do this, is to make sure that when we have a piece spin
There’s room for it to spin
Otherwise, you have it hit other things
And maybe break
And then bad things happen.
So, let’s look at a few examples that you can expect for homework
EXAMPLE 2:
Imagine you have a square
And you spin it continuously around the x-axis
How would you describe the shape it creates?
Okay, to start off with, we need a square:
Now, we’re going to spin this square on it’s x-axis
Like so:
Well, if we spin it around, we can see that we’ll get something circular on top
Something circular on the bottom
And, a square (rectangle) that is the diameter of the circle times pi
What was that again?
Oh yeah
A CYLINDER.
Example 3:
Imagine you have a right triangle
And you spin it continuously around the x-axis
How would you describe the shape it creates?
Okay, to start off with, we need a right triangle:
Now, we’re going to spin this right triangle on it’s x-axis
Like so:
Well, if we spin it around, we can see that we’ll get something circular on the bottom
And, it seems to go to a point on top
This is a new one, because it’s not really a cylinder
But a pointy cylinder is called a
CONE
Example 4:
Imagine you have an isosceles triangle
And you spin it continuously around the y-axis
How would you describe the shape it creates?
Okay, to start off with, we need an isosceles triangle:
Now, we’re going to spin this triangle on it’s y-axis
Like so:
Well, if we spin it around, this is what we see at first:
But, if we look at the base, what we see instead is:
Now, this is where it gets weird because if we look at just the left triangle:
We can see it’s just a cone.
And if we look at just the right triangle:
We can see it’s also just a cone.
So, what we have here are two cones sharing a base: