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Area of Parallelograms - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The area of parallelograms formula is derived from the area of a rectangle. By drawing an altitude from one vertex in a parallelogram and translating the right triangle, a rectangle is formed. Therefore, to calculate the area of a parallelogram, multiply a height by the corresponding base. The corresponding base is the side perpendicular to the height. Related topics include area of trapezoids and rhombuses.

When we talk about area, a rectangle doesn't really have an area. What we're talking about is the region bound by the rectangle. So this rectangle doesn't really have an area but this region that is formed by the sides of the rectangle has an area.
We go back to grade school and I asked you what was the area of this rectangle? You'd say, "Mr. McCall, the area is equal to length times width." Well, now that we're in Geometry, we're going to use different language. Instead of length and width, we're going to use base times height. So forget about length times width and we're going to use the base and the height. So the area of a rectangle is going to be your base, length times its height. But how does that apply for a parallelogram that's not a rectangle?
Well, what we could do, is if I had a scissors strong enough to cut through this white board, I would cut off this triangle right here and I would translate it in that direction and I would paste it back on right on that end. So what I've created here is a rectangle out of my parallelogram. So right here, we would have that little triangle that I cut off at that end. Notice that we've created something that we know the area, a rectangle. So we can say the area of any parallelogram will be equal to its base times its corresponding height.
Now that's a little tricky [IB] what do I mean by corresponding height? Well, let's say I determine that this right here was teh base of my parallelogram. So I'm just assuming that we have 2 pairs of parallel sides here. If this is my base, the coresponding height is a perpendicular segment that connects its opposite base. So this right here could be a height because, it starts at its parallel base and it's perpendicular to b.
You could also have a height that is outside of your parallelogram. Let's say okay, taking the exact same figure. If I said that this right here was my base. Then what would I need to know in order to calculate the area? Well here, my corresponding height would still be a perpendicular segment which could be inside my parallelogram or if I extended this side it would look something like this.
So remember, area equals base times height where your height is a perpendicular segment to your base. So that's what you have to know in order to calculate the area of any parallelogram. So this applies to rectangles, parallelograms, rhombuses and squares.

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