Volume of Cones
Objective
Volume of Cones
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.
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, now let’s look at the volume of pyramids:
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:
Now that we know how to find it
Let’s look at some examples:
Example 1:
Find the volume from the following:
453’
58’
So, we’re trying to figure out the volume for the Great Pyramid of Giza
And we know that the formula for the volume of a pyramid is:
So:
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
So, let’s start off with a few examples!
Example 1:
Find the volume from the following:
Now let’s look at the volume of the terrifying tornado
So we know that the formula for the volume of a cone is:
So:
20’
200’
EXAMPLE 2:
4”
10”
Find the volume from the following:
So we know that the formula for the volume of a cone is:
So for the top cone, it’ll be:
9”
And we know that we have two cones here
And for the bottom cone:
Now we add those two together, and we get: