12 Area
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John is a painter who is going to paint a wall in a house. What he needs to know is how much paint he needs to buy for this one particular wall. He knows that the wall is in the shape of a rectangle and that the dimensions are 10 feet high by 27 feet wide.
How do you think he would go about solving this problem?
The answer lies in finding the area of the wall. A can of paint will be able to cover a certain area and if John can find out the area of the wall then he can figure out how many cans of paint he will need.
The first thing we should do here is write down the definition of area.
Area: The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle, square, or circle.
The following are some two dimensional shapes shaded in grey. The grey portion represents the area of the object while the black lines surrounding the object represent the perimeter.
Area of a Square or Rectangle
Before we begin going through the motions to calculate the area of a square or rectangle lets take a quick visit back to perimeter and more specifically how we defined the dimensions of both a square and a rectangle.
Once again the portion shaded grey is the area of each of the object. The question becomes this…
Can we use those dimensions when calculating the area? Or do we need to somehow come up with other dimensions in order to get to our answer?
Well as it turns out, those dimensions will work for us. Not only do they come in handy when we are calculating perimeter, but they also work really well when calculating area.
The next question then becomes HOW do we use these dimensions?
Before you move on to see how it’s done, take a minute to think about it. Maybe even write down some of your thoughts. Once again, this goes back to something we talked about before. If you are able to understand the concept then the process involves less memorization.
Here are the formulas for calculating the area of both a square and a rectangle.
[latex]\Large \begin{array}{ll}\textbf{Square:} & \text{Area} = \text{side} \times \text{side}\\ \textbf{Rectangle:}&\text{Area}= \text{length}\times \text{width}\end{array}[/latex]
Take note here as there are a couple things different than dealing with perimeter. The first is that we are now multiplying rather than adding. This ends up leading to our second point.
If we take a look at the formula for the square we see that we multiply a side by a side. As all the sides are the same it doesn’t really matter which two sides we multiply together.
The issue becomes the units that we end up with. Remember that when dealing with perimeter we are only dealing with a one dimensional line. Our units end up being linear or essentially one dimensional.
With area we end up with units that give us an answer using two dimensional units. The best way to understand this is to go through an example.
Let’s say we have a square where each side