Hi! Welcome to Math Antics.
In our last video, we learned about circles and we learned about a special ratio called Pi.
In this video, we’re going to learn how we can use that ratio to calculate the circumference and the area of any circle.
The formulas that we use to calculate circumference and area are so important that you should really memorize them.
To help you do that, we're going to look at them side-by-side.
That will help you see their similarities and their differences so you don't get them mixed up.
The formula for finding the circumference is:
Circumference equals Pi times diameter.
And just like most formulas, we use abbreviation:
'C' for circumference and 'd' for diameter.
That's a pretty simple formula.
It tells us that if we know the diameter of a circle, all we have to do is
multiply that diameter times the number Pi and we'll get the circumference.
We'll try that formula out in a few minutes.
But first, let's see the formula for area.
The formula for finding the area of a circle is:
Area equals Pi times radius squared.
Again, we can use abbreviations to make it shorter:
'A' for area and 'r' for radius.
Now this is a pretty simple formula too.
It tells us that if we know the radius,
we just have to 'square' it and then multiply that times Pi to get the area.
Okay, but what does it mean to 'square' the radius?
Well, squaring a number just means multiplying it by itself.
For example, 3 squared just means 3 times 3,
and 5 squared just means 5 times 5
and r squared just means r times r.
So our formula is really just:
Area equals Pi times r times r, but we write it in the 'r squared' form because it's more compact.
Oh, and one really important thing to keep in mind is that
r squared is NOT the same thing as 2 times r.
That's a common mistake that students make when first learning how to find the area of a circle.
And if we look carefully at both of our formulas, you'll see why.
These two formulas have a lot in common.
In each of them, you are multiplying Pi by part of a circle to find either the circumference or the area.
In the case of the circumference, you are multiplying Pi times the diameter,
and in the case of area, you are multiplying Pi times the radius squared.
But do you remember the relationship between the radius and the diameter?
Diameter is just 2 times the radius.
So we could re-write our formula for circumference like this:
Circumference = Pi × 2 × r.
Ha! Now you see why it's so easy to get confused.
To find the circumference, you take the radius and double it.
Then you multiply by Pi to get the final answer.
But for area, you don't double the radius… you square it.
That's a very important difference.
To help you see that difference in action,
let's find both the circumference and the area of this circle using our two formulas.
The only thing we know about this circle is that the radius is 8 meters.
Luckily, that's all we need to know.
First, we use our formula for circumference: C = Pi × d.
To get the diameter, we take the radius and we double it. …that is, we multiply it by 2.
2 × 8 = 16, so the diameter is 16 meters.
Then, we multiply that by Pi to get the circumference.
Since this is decimal multiplication, I'm going to use a calculator.
16 × 3.14 = 50.24
So that means that the circumference of this circle is 50.24 meters.
Alright, now let's find the area using our formula: A = Pi times r squared.
Again, we start with the radius, but instead of doubling it, we 'square' it.
That means we multiply it by itself.
8 m × 8 m = 64 meters squared.
Then we multiply that by Pi.
64 × 3.14 = 200.96 meters squared.
That's the area of this circle.
As you can see, the result we get when we square the radius
is very different from the result we get when we we double it.
And one of the most important differences is with the units of our answer.
Doubling the radius just gives us the diameter, which is a 1-dimensional quantity.
So, the answer we get from our formula for circumference is also a 1-dimensional quantity.
But, when we square the radius, that gives us 'square units', which are 2-dimensional.
That makes sense because area is always a 2-dimensional quantity.
Remembering that will help you avoid getting these two formulas mixed up.
The one that has the radius squared is always for area.
Alright, let's try a couple real-world examples to make sure you've got it.
Here's the real world, which as you probably know is a sphere.
But, if we take a slice of the world, right at the equator, that slice is a circle.
Let's find the circumference of that circle.
To do that, we need to know the diameter of the earth.
That turns out to be about 12,750 km.
Great, then to find the circumference we just need to multiply that diameter times Pi.
Now I'm definitely going to use a calculator for this.
And, I'm going to use a more accurate version of Pi since this is such a big distance.
12,750 × 3.14159 = 40,055 km (to the nearest kilometer).
Wow, that's a pretty big circumference!
No wonder it takes so long to go all the way around the earth!
On your mark… Get set… Go!
Whooo - Yes! 3.14 seconds quicker than last time. Yes!
Here's another real-world example with a circle.
If this pizza has a diameter of 24 inches, what's its total area?
Well, using our formula, we start by squaring the radius.
But, the problem didn’t give us the radius…
it gave us the diameter, so we have to calculate the radius from the diameter.
Fortunately, that's really easy.
The radius is just half of the diameter, so we just need to divide the diameter by 2.
24 inches divided by 2 gives us 12 inches for the radius.
And now that we know the radius, we need to square it.
12 in × 12 in = 144 inches squared.
Next, we just multiply that by Pi.
144 × 3.14 is 452.16.
So, the total area of the pizza is 452.16 square inches.
Alright, so know you know how to find the circumference and the area of any circle.
All you need to do is remember the formulas:
Circumference equals Pi times diameter,
and Area equals Pi times radius squared.
But, it's really important to practice using these formulas for yourself,
so be sure to try some of the exercises problems.
That's the way to really learn math.
Thanks for watching Math Antics and I'll see ya next time.
Learn more at www.mathantics.com