Source: All images created by Anthony Varela
Hi, and welcome. My name is Anthony Varela, and today I'd like to talk about area. So we're going to define area-- talk about what it is. Then I'm going to introduce some formulas for the areas of different shapes, and we're going to use those formulas to solve problems involving area.
So to get started, what is area? Well, perhaps you've seen blueprints or a floor plan before. And this just puts the floor of a home in 2-dimensional space. And someone interested in this house might wonder, what's the area of the master bedroom? And they're talking, then, about the 2-dimensional space inside one of the rooms.
So area is the space enclosed in 2-dimensional objects. So we're talking about 2D here. So units to measure area, then, are squared units. And some examples of squared units are meters squared or squared feet. There's, of course, others.
And now, let's talk about the formulas for some areas of different shapes. So for rectangles, the area is base times height. And for circles, the area is pi r squared, where r is the radius and pi is a number-- approximately 3.14.
So now, let's calculate area. So here, we're looking at the side of my house. There's some pavement there to park my car. And I know the dimensions of this rectangular parking space are 2.4 meters by 4.5 meters. And I'd like to know the area of this parking space.
Well, I have a rectangular shape, so I'm going to pull up my formula for the area of a rectangle, A equals bh-- base times height. And so now, I'm just going to substitute 2.4 and 4.5 in for my base and my height. So I have A equals 2.4 meters times 4.5 meters.
Now, just multiplying those two dimensions together, I get 10.8. And I know that my unit's going to be meters squared, because area is measured in square units. I can also see that here is m times m equals m squared.
Well, let's look deeper into my backyard. And I have a circular pool. And there's a tarp that goes over the pool to protect it in the off season. And I know that the radius of that tarp is 7 and 1/2 feet. What is the area of this circular tarp?
Now, before we get started, I want to mention, sometimes, you're only given the diameter of a circle and you're asked to find the area. Well, the radius is 1/2 of the diameter. So you can just divide that diameter by 2 if you're given diameter and need to find the radius.
Well, we're going to pull out our area for a circle-- pi r squared. And I know that my radius, r, is 7 and 1/2 feet. So now I have to square that radius, so 7 and 1/2 feet squared. This gives me 56.25.
And I'm multiplying that by pi. And then my units went from feet to feet squared. Well, pi is approximately 3.14, so I can multiply 56.25 by 3.14. And so this area, then, is about 176.63 squared feet.
Well, now let's solve some area problems a bit differently. Here, I happen to know what the area of my parking space is. It's 13.14 squared meters. And I know one of the dimensions, but I don't know this dimension here.
So I'm going to use what I know about area so far-- A equals b times h. And given the area as 13.14 meters squared, and that equals the base, 1.8 meters, times the height, which I don't know. So what I need to do now is divide both sides of this equation by 1.8 meters.
That will give me my x value on one side of the equation, and 7.3 meters on the other. Notice that meters squared divided by meters gives me meters. So this length here is 7.3 meters.
Let's return, then, to my swimming pool-- the tarp cover that goes over the swimming pool. What if I know that the area is 214 squared feet and I want to find the radius? Another side note here. If you want to find the diameter, we can solve for this radius and then just double it to find the diameter.
Well, let's start with our formula for the area of a circle, A equals pi r squared. And I know that the area is 214 square feet. So that equals pi r squared. Well, what I can do, knowing that pi equals approximately 3.14, is I'm going to divide both sides of my equation by pi.
So 214 divided by pi is 68.15. And then pi r squared divided by pi is just r squared. Notice so far, nothing has happened to my units, because pi is just a number. It's not a unit.
So now, how do I find just the radius? I have an expression for the radius squared, so what I have to do is take the square root of both sides of my equation. So the square root of r squared will bring me my radius.
And then the square root of 68.15 squared feet is 8.26 feet. Notice that taking the square root of square feet brings me back just regular old linear feet. So the radius of this tarp, then, is 8.26 feet.
So let's review. Today, we talked about area being the space enclosed in 2-dimensional objects, and it's measured in square units. Some commons units are meters squared or squared feet, for example. We used the formulas for the area of a rectangle and the area of a circle to solve some problems.
And remember that pi is the ratio of a circle's circumference to its diameter, approximately 3.14. Well, thanks for watching this tutorial on area. Hope to catch you next time.