Volume of a Sphere

Objective

• Review over the volume of a pyramid
• Review over the volume of a cone
• Review over the surface area of a sphere
• Go over the volume of a sphere
• Do some examples
• Homework

SO HOW DO WE FIND THE VOLUME OF A PYRAMID

Well, to find the volume of anything, let’s first look at the net.

Now we know that a triangle is half a square

So, looking at this net, it would make sense that a pyramid would also be half of a prism, right?

Except let’s take this and really look at it:

As we can see, when we take the sides of a pyramid

And put them next to each other, we only make 2 sides of a prism

And we know that a prism actually has:

6 sides

So, since a pyramid only makes 2 sides, then we can see that instead of being half of a prism

It’s actually one third of a prism.

So, since we know the volume of a prism is:

(height of the prism)*(area of the base)

And a pyramid is one third of a prism

Then the volume of a pyramid is:

1/3 *(height of the pyramid)*(area of the base)

SO, DOES THIS ACTUALLY WORK?

It actually does!

Here is a physical example:

So, the formula for the volume of a pyramid is:

So, now let’s review over conic volume:

OKAY SO WHAT ABOUT CONIC VOLUME?

Well, remember when we went over surface area for cones?

And we realized that cones are just pyramids with a circle for a base?

Well, that’s really the same thing here.

When we’re talking about the volume of a cone, we’re talking about the volume of a pyramid

Just with a circle for a base

So the equation is still the same

But with a slight twist.

Let’s look at the equation for the volume of a pyramid:

Now, when we’re dealing with a cone, we know the base of the cone is a circle

And the area of a circle is actually pretty easy to find:

So, if we substitute the area of a circle in with the area of the base

Then we get:

And now we have the equation for the volume of a cone

Now let’s go over the surface area of a sphere

NOW LET’S TAKE A LOOK AT THE SURFACE AREA OF A SPHERE.

Alright, now before we start talking about the surface area of a sphere

Let’s look at one:

Normally, we’d talk about how to find the lateral area

Then use that to find the surface area

Then go from there

But, what exactly is the lateral area of a sphere?

Since it’s sort of missing any sort of base

It’s hard to find

So instead, let’s look at something that’s more like a sphere

Like a cylinder

Comparing Spheres to Cylinders

Just so you are aware, we’re going to use Archimedes’ proof to show this.

So, first we need our sphere:

And we need our cylinder:

Now, let’s say our sphere has some radius, right?

r

And let’s say our cylinder has the exact same radius.

r

Since we’re talking about our cylinder, let’s say it’s height is equal to 2r (or the diameter of the sphere)

2r

It would seem that the cylinder is the same height as the sphere

And it’s the same width as the sphere

So, if we place the sphere inside the cylinder:

Then we can see that the surface area of the cylinder is actually the same as the spheres

Basically imagine you’re unfolding the sphere into a cylinder

Then you’d have the cylinder that we have right now.

Okay, so what?

Normally we wouldn’t care, but since we now can see the surface area of the sphere is the same as the lateral surface area of the cylinder

We can use the lateral surface area of a cylinder to find the surface area of a sphere.

So, remember, the lateral surface area of a cylinder is:

And since we made the height of the cylinder

2 * the radius

That leaves the circumference of the circle, which is:

And since we now know that the lateral area of a cylinder is the exact same as the surface area of a sphere

Then the formula for the surface area of a sphere must be:

= Surface Area of a Sphere

And that’s it!

That is the surface area of a sphere

So, of course, this got a little weird, but basically

The surface area of the sphere is 4 times the are of a circle.

Now let’s talk about volume

GETTING INTO THE RIGHT MINDSET….

So, just like when we found the formula for the surface area of a sphere, and we needed to think outside of the box,

Well, we’re going to need to think differently with volume as well.

So, just like before, we’re going to need to use some other shapes to kind of force the equation out.

However, instead of using cylinders with spheres this time

We’re going to use cones and spheres

I know this sounds weird, but it’ll make more sense when we go through it:

DERIVING THE FORMULA….

So, to start out with, we need a sphere:

Now, what we’re going to do is take two cones

And place them in the sphere, one on top of the other:

Now, we can see that these cones share the same radius as the sphere

They’re also as tall as the radius (since they start at the middle of the sphere, and take up half of the sphere)

r

So, now that we have these two cones inside the sphere

We can use the volume formula to find the volume of these two cones

So:

Which will give us:

But, what we have is only in two dimensions!

We haven’t even considered a whole other side.

So, let’s rotate the sphere with the cones:

And let’s look at our other two cones:

Now, just like the other two cones inside the sphere

We can use the volume formula to find the volume of these two cones

And doing so would get us another:

So, adding these two together gets us:

Which simplifies to:

AND THAT’S IT!

That’s how you find the volume of a sphere

You take the radius

Cube it

And multiply it by 4/3 pi

So now that we know how to do it

Let’s look at some examples:

EXAMPLE 1:

Find the volume of Jupiter:

Well, we know that the volume formula is

And we know the radius of Jupiter

So:

43,441 miles

Example 2:

Find the volume of the marble:

Well, we know that the volume formula is

And we know the radius of the marble

So:

1”

Example 3:

Find the volume of the Sun:

Well, we know that the volume formula is

And we know the radius of the sun

So:

432,690 miles