In this video, we're going to talk about how to calculate the area and the circumference of a circle. But first, let's talk about the radius and the diameter of a circle. My drawing's not perfect, but we're going to make the best of that. So here is the center of the circle.
It's a point right at the middle of the circle. Now the distance between the center of the circle and any point on the circle, that is known as the radius of the circle. The diameter is twice the value of the radius. So let's say this is the center of the circle. The diameter starts from one end of the circle, it passes through the center, and connects to the other end of the circle.
So as you can see visually, it's twice as long as the radius of the circle. So we can say that D is equal to 2R. Now the next thing you need to be familiar with is the area and the circumference of a circle.
The area is basically the area of the shaded region. To calculate the area of a circle, you could use this formula. It's pi times r squared. Now the circumference is basically the distance around the edge of the circle. So this is the circumference.
And to calculate the circumference, it's equal to 2 pi times r. Now, pi is equal to 3.141592654. And there are more other numbers, you know, that go on after that. But typically, when solving these problems in school, you'll be using approximate value of pi, which is usually 3.14.
So now let's work on some problems. Let's start with this one. Number one, calculate the area and circumference of the circle shown below.
Use pi equal 3.14. So let's start with the area. The area is equal to pi r squared. The radius, we can clearly see that the radius of this circle is 5 feet. It's the distance between the center and the point on the circle.
So we have pi times 5 feet squared. 5 squared is 5 times 5, which is 25. And feet times feet gives you square feet. So the units of area is always square units. Square inches, square feet, square yards, and so forth.
Now, the exact answer for the area of the circle is 25 pi square feet. But we're going to get an approximation using this value. So let's replace pi with 3.14. So the answer for this problem, the area is approximately 78.5 square feet. So that's how you can calculate the area of a circle if you're given the radius of that circle.
Now let's move on to the second part of the problem. Let's calculate the circumference of the circle. So we're going to use the formula C is equal to 2 pi r.
Let's get the exact value first. So let's replace the radius with 5 feet. 2 pi times 5 is 10 pi. So the exact answer is 10 pi feet. So notice that the unit for circumference is in feet, which is the same as the unit for radius, but the unit for area is in square feet.
Now let's replace pi with 3.14. So this will give us an approximate answer. 10 times 3.14 is going to be 31.4. The circumference of this particular circle is 31.4 feet.
So if you were to start here, let's say on a track, and let's say you were to walk around this circle, you would travel a distance of 31.4 feet. So that's what that means, given that the radius of the circle is 5 feet. Now let's work on a similar problem.
This time we're given the diameter of the circle. The diameter is 14 inches. How can we use the diameter to calculate the area and the circumference of the circle? Well the first thing we need to do is we need to find the radius. Remember the diameter is twice the length of the radius.
So if we divide both sides of this equation by 2, we'll get that the radius is half of the diameter. Half of 14 or 14 divided by 2 is 7. So the radius of this circle is 7 inches. So now that we know the radius of the circle, we can calculate the area and the circumference. So let's start with the area.
We know the area is pi r squared. So let's replace r with 7 inches. 7 squared or 7 times 7 is 49. So we get that the area is 49 pi square inches.
So this right here is the exact answer. But now, let's replace pi with 3.14. 49 times 3.14, that's 153.86.
But what we're going to do is we're going to round our answer to 154 square inches. So that's the area of the circle in this problem. Now let's calculate the circumference. The circumference is 2 pi r.
Let's replace r with 7 inches. So we have 2 pi times 7. So we get an exact answer of 14 pi inches. Now let's get our approximate answer by replacing pi with 3.14.
14 times 3.14, that's 43.96. But since we used 3.14 for pi, I'd like to round my answer to three significant figures. So I'm going to round it to 44.0 inches.
So that's the approximate value for the circumference of the circle. Number two, the area of a circle is 28.5 square inches. What is the radius of the circle? So we're going to use 3.14 for pi, and we're going to round our answer to the nearest hundredth place.
So let's start with this formula. Area is equal to pi r squared. So we're going to replace the area with 28.5 square inches.
And we're going to replace pi with 3.14. And we're going to solve for r. In order to do that, we need to divide both sides by 3.14.
So we get 28.5 divided by 3.14, and that works out to be 9.076433, with some other numbers. But this should be good enough. So that is equal to r squared.
To get r, we need to take the square root of both sides. So the square root of 9.076433. We could round that answer. So let's use our approximate symbol.
So we're going to round it to 3.01. So that's the nearest hundredth. So this is the radius of the circle. That's how we could find it if we're given the area of the circle.
And by the way, let's not forget the unit for radius. The unit for area is square inches. So the unit for the radius of this circle is going to be just inches. The radius, diameter, circumference, they have units like inches, feet, yards, things like that.
Area is always square units. So just keep that in mind. But these two, they must match.
Number three, the circumference of a circle is 14.5 feet. What is the diameter of the circle? So the formula that we need to use is circumference is equal to 2 pi r.
Now what we need to do here is we need to use the circumference to calculate the radius of the circle. Once we have the radius of the circle, we can then find the diameter of the circle because we know that the diameter is twice the value of the radius of the circle. So let's calculate r first. Let's replace c with 14.5 feet.
And let's replace pi with 3.14. 2 times 3.14, that's 6.28. So to get r by itself, we need to divide both sides by 6.28.
14.5 divided by 6.28. We get 2.3089. And we're going to round it to the nearest hundredth. So that's going to be 2.31. Now let's put the units.
So the unit for circumference is in feet. That's going to be the same for the unit of radius. So now that we have the radius, we can find the diameter. The diameter is simply 2r. It's twice the value of r.
So it's going to be 2 times 2.31. 2 times 2 is 4. 2 times 31 is 62. So the diameter of this circle is going to be approximately 4.62 feet. Number 4. The circumference of a circle is 18 pi meters. What is the area of the circle? Go ahead and try that problem.
So let's begin with this equation, c is equal to 2 pi r. So that's our circumference equation, and we also know that the area is pi r squared. So those are the two equations that we have.
So we're given the circumference, and we need to find the area. What we need to do is we need to use this. circumference to get the radius.
Once we have the radius, we could plug it into the second equation to get the area. So let's go ahead and do that. So we're going to replace the circumference with what it equals, 18 pi meters.
Now we're going to solve for r. To get r by itself, we need to divide both sides by 2 pi. On the right side, 2 pi will cancel. On the left side, pi will cancel. And we're going to have 18 divided by 2, which is 9. So this is the radius of the circle.
It's 9. And since the unit for circumference is in meters, the radius is going to be in meters as well. So this is an exact answer. So now let's use that to calculate the area. So let's replace r with 9 meters.
9 squared is 81. So the exact answer for the area will be 81 pi square meters. Now let's get our approximate answer. Let's replace pi with 3.14. So 81 times 3.14, that's 254.34.
So we're going to round that to the nearest whole number, which will be 254. So that's the approximate area of the circle. It's 254 square meters. That's the answer.
Number five. The area of a circle is 100 square yards. What is the circumference of the circle? So let's write the two equations that we're going to use. is equal to pi r squared and C is equal to 2 pi r.
So this problem is basically the reverse of the previous problem. We're given the area, we're going to use that to calculate r. Once we have r, we're going to plug it into this equation to get the circumference. So let's replace a with 100 square yards. And let's plug in pi as 3.14.
So to get r by itself, what we're going to do is we're going to divide both sides by 3.14. 100 divided by 3.14. And that is 31.847133. And that's square yards. So to get r, we need to take the square root of both sides.
So the square root of 31.847133, that's 5.6433. 3. And the square root of yards squared is just yards. So this is the approximate value for the radius. So now we're going to take that answer and plug it in to the circumference equation.
So we have c is equal to 2 pi, but we're going to use 3.14 for pi, and then we're going to replace r with 5.64333 yards. So you should get 35.440, but we're going to round it to the nearest tenth, so we're going to say that the circumference is approximately 35.4, and the unit is going to be yards. So that's how you can calculate the circumference if you know the area of the circle.
And that's basically it for this video. Thanks for watching.