It’s Professor Dave, let’s talk about 3D shapes.
For quite a few tutorials we’ve been looking at lines and polygons and circles, and all
kinds of other things, but these geometrical figures have all been contained in a single plane.
They are all two-dimensional.
We know that in the real world, objects are three-dimensional, nothing is completely flat,
not even a piece of paper, so it’s important to understand the geometry of three-dimensional
shapes as well.
Let’s go through some of these shapes now, and talk about some of their parameters that
we can calculate.
First of all, rather than calling these things shapes, let’s refer to them by their proper
name, polyhedrons.
A polyhedron is a three-dimensional object with flat surfaces called faces, and each
face of a polyhedron is a polygon.
Where two faces meet and share a side, this is called an edge, and any point where three
or more edges meet is called a vertex.
The first type of polyhedron we will look at is the prism.
This is a shape with two congruent parallel bases of some type of polygon, and all the
other shapes are parallelograms.
A prism is named for the shape of its bases.
This is a rectangular prism.
We can see the bases, and all the rectangular sides, which qualify as parallelograms.
We can have pentagonal and hexagonal prisms as well, we just have parallel bases of the
shape that is referenced, and then as many rectangles connecting the two bases as are
necessary.
Another type of shape is a cylinder.
Cylinders have two circular bases, but they are not considered polyhedra, because none
of these surfaces are polygons.
Next we want to look at pyramids.
Unlike prisms, these have only one base, which is a polygon of some sort, and rather than
parallelograms that connect to another base, we have triangles that come together at a point.
A pyramid is named for the shape of its base.
This is a square pyramid.
This is a triangular pyramid.
And we could certainly have pyramid bases with more sides than this.
A similar shape is a cone, which is sort of like a pyramid but with a circular base.
Again, cones are not polyhedra, because these faces are not polygons.
Now, what can we do with these polyhedra, what can we describe?
Well we know how to calculate the area of certain polygons, so we could calculate the
total area of all the faces of a polyhedron.
This is called the surface area, quite literally the area on the surface of a three-dimensional
shape, which we can think of as the amount of wrapping paper we would need to wrap up
a gift of a particular shape, just barely covering all of it.
Take something like a rectangular prism.
This has six sides.
These two are the same, these two are the same, and these two are the same.
To get the surface area, we just have to get the area of each rectangle, and add them all up.
Given the dimensions of two by three by four meters, we get two rectangles that are two
by three, two that are two by four, and two that are three by four.
That’s two times six plus two times eight plus two times twelve, which equals fifty-two
square meters.
What about for a pyramid like this square pyramid?
Now we have a couple different shapes.
We have this square base and then four triangles, so it’s the formula for the area of a square
plus four times the area for a triangle.
If we have a base with a side length of five meters and triangles with bases that are just
the side of the square, and with heights of six meters, then we can get the area of the
square first, which is five squared, or twenty-five, and then each triangle will be one half five
times six, or fifteen.
There’s four of them, so that’s sixty.
Twenty-five plus sixty is eighty-five square meters.
What about a cylinder?
This one seems like it would be complicated, but it’s actually just two circles plus
a rectangle that’s curved around on itself.
We can imagine slicing through the height of the cylinder and unwrapping it to get this
rectangle, which will have one side equal to the height of the cylinder, and another
side equal to the circumference of the base of the cylinder.
So the area of a cylinder is twice the area of the circle, since there’s one on the
top and one on the bottom, plus the area of this rectangle, which is height times circumference,
which is pi D or two pi R. So if we have a radius of two meters and a height of five
meters, we can get the rectangle first, which will be five times two pi times two, or twenty pi.
Then we have two circles, which means pi times two squared to get the area for each circle,
which will add up to eight pi.
Add these values together and we get twenty eight pi.
Lastly, let’s find the surface area of a sphere, which is the set of all points in
space that are a fixed distance from a central point, with that distance being the radius
of the sphere.
The surface area of a sphere is given by four pi R squared, so with a radius of five, that
would be twenty-five, times four is one hundred, times pi is one hundred pi square meters.
That’s quite a lot of shapes and ways to calculate surface area, so let’s check comprehension.