Surface Area of a Sphere
Objective
So, let’s start first with a regular pyramid
To start, a regular pyramid has a regular polygon as a base.
Which means all of the sides of the base are the same size.
Now, to start, we need a pyramid to look at.
Something like this:
So, to find the surface area of this pyramid, we need to find the area of
This triangle:
And this triangle:
And this triangle:
And this triangle:
And then we’ll just add all of those areas together to get the entire surface area of the object.
Seems pretty simple right?
Except we know that finding the area of all of those triangles is anything but simple.
So how can we come up with a better way to find the surface area?
Well, we may need to look at the net again.
And of course, we still need the area of the base:
Looking at the net
Remember, we’re starting with this pyramid here:
Now, let’s give it some measurements:
9”
12”
And take a look at its net:
Now, as we can see from the net, the base of each triangle is equal to a side of the base:
And each triangle has a height of the pyramid as well:
So, to start, let’s find the area of each triangle.
So:
However, that’s a really complicated way of looking at it
What if we factored this?
Well, then, if we factored out the 1/2, it would look like:
But wait, if we factor out the height as well, we get:
And isn’t that just the perimeter of the base?
So then, we can say that the lateral area of the pyramid is going to be:
9”
9”
9”
9”
12”
AND THAT’S HOW YOU FIND THE LATERAL AREA OF A PYRAMID
So the lateral area (area without the base) of a pyramid 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 pyramid only has one base
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 pyramid, it would be:
9”
9”
9”
9”
12”
AND THAT’S IT!
Just like any of the other surface areas we’ve dealt with before
To find the surface area of a pyramid you need to:
- Find the lateral area
- Then add the area base.
So, let’s take a look at an example:
EXAMPLE 1:
Find the surface area:
4”
6”
Remembering the formula:
First we need the Lateral Area
So, plugging in the numbers for the lateral area formula
And the perimeter of the base is:
4 + 4 + 4 + 4
= 16
Then we get:
Lateral area = 48
Now we just need the area of the base
But we know that’s going to be:
= 16
Then we can plug this into our Surface Area formula and get:
So our surface area for this prism is: 64 square inches
NOW LET’S LOOK AT CONIC SURFACE AREA
To start out, we’re going to need a right angle cone
Something like this:
Now, to start, we need a pyramid to look at.
Something like this:
So, to find the surface area of this pyramid, we need to find the area of
This piece:
And then we’ll just add all of those areas together to get the entire surface area of the object.
Seems pretty simple right?
Except, how do we find the area of the top piece?
Well, let’s really look at this:
And of course, we still need the area of the base:
Looking at the cone
Remember, we’re starting with this cone here:
Now, let’s give it some measurements:
4”
10”
Now, this cone looks an awful lot like our pyramids, right?
And we know that the lateral area of a pyramid is:
And since we know that the perimeter of a circle is actually the circumference of a circle, which is:
Then, if we plug this into our equation for a pyramid, we get something that looks like:
Which, when multiplied out, actually becomes:
And that’s how you find the lateral area of a cone
So the lateral area (area without the base) of a pyramid is always going to be:
Now the question becomes
How do we find the total surface area?
Well, we just need to add the area of the base right?
So, looking at our cone:
We can see the base is just a circle.
So, we can add the area of the circle to the lateral area and get what we’ve been looking for!
So:
4”
10”
So for this cone, the formula would look like:
And that’s it!
Just like any of the other surface areas we’ve dealt with before
To find the surface area of a cone you need to:
- Find the lateral area
- Then add the area base.
So, let’s take a look at an example:
Example 1:
Find the surface area of this tornado:
20’
200’
Remembering the formula:
First we need the Lateral Area
So, plugging in the numbers for the lateral area formula we get:
Now we just need the area of the base
But we know that’s going to be:
Then we can plug this into our Surface Area formula and get:
So our surface area for this prism is approximately: 13823 square feet
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.
So, let’s see some examples:
EXAMPLE 1:
Find the surface area of Jupiter:
Well, we know that the surface area formula is
And we know the radius of Jupiter
So:
43,441 miles
EXAMPLE 2:
Find the surface area of the marble:
Well, we know that the surface area formula is
And we know the radius of the marble
So:
1”
Example 3:
Find the surface area of the Sun:
Well, we know that the surface area formula is
And we know the radius of the sun
So:
432,690 miles