Surface Area of Pyramids and Cones
Objective
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:
EXAMPLE 1:
Find the surface area:
5”
5”
18”
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:
18 + 18 + 5 + 5
= 46
Then we get:
Lateral area = 230
Now we just need the area of the base
But we know that’s going to be:
= 90
Then we can plug this into our Surface Area formula and get:
So our surface area for this prism is: 410 square inches
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:
Example 1:
Find the surface area:
6”
15”
Remembering the formula:
First we need the Lateral Area
So, plugging in the numbers for the lateral area formula:
Now we just need the area of the base
But we know that’s going to be:
So, plugging these into our formula, we get:
Which is approximately 565.49 inches
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