Area of Polygons

This video gives formulas and examples for how to find the area of squares, rectangles, triangles, parallelograms, and trapezoids. The video also explains the difference between base and height.

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transcript:

Welcome to Your Tutor Online video podcast. In this lesson, we’re going to look at area for several different polygons. Area is a measurement of a region that a shape takes up. It’s measured in units squared. The easiest way to find area is just to count the number of squares that it takes up on graph paper. Look at the blue box I just drew and we can easily see the area by counting the boxes -- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So this square has an area of 12 units squared.

Sometimes we can’t just count the squares on graph paper because it’s unavailable, so we need to learn the following formulas for area. The first and the easiest shape we’re going to look at are squares and rectangles. Squares are rectangles so they have the same area formula. Area is equal to the length times the width. Or in other words, just one side times the other side.

Here’s a very easy example. We have a rectangle with a length of 7 inches and a width of 3 inches. 7 times 3 is 21. We’re going to write the same units that we see, inches, and then we’re going to put a little 2 up above the inches to show that’s inches squared. And that way, everybody knows we’re talking about the area and not just the length.

The next shape we’re going to look at is a triangle. The formula for the area of a triangle is area is equal to one-half times the base, times the height. The base is just going to be any of the sides of the triangle, and the height extends from that base to the opposite vertex at a perpendicular angle.

Let’s look at base and height a little more closely because I know it can be confusing.
First, for a right triangle, we’re just going to pick the two legs of the right triangle for your base and height. It doesn’t matter which one because multiplication can be done in any order.

For other triangles, it gets a little more confusing. I normally just choose the side that’s closest to me to be my base, and then the height is going to go from the base to the opposite vertex, down exactly at a 90° angle.

For the flask triangle, again, I’m going to choose the side closest to me for my base. The opposite vertex is all the way over here. Drop directly down from the vertex to my base. And notice here, the height is actually outside of the triangle, which is OK.

Let’s look at an example. Now, the hardest part of figuring out the area of a triangle is to know which one is your base, and which number is your height. So we have all these numbers here, our clue is with the right angle symbol. The right angle symbol is going to join together your base and your height. So we see this side, 4 centimeters, and this side, 3 centimeters, as our base and our height. Just to refresh your memory on the area, it’s equal to one-half base times the height. 4 times 3 is 12, times one-half, is equal to 6 centimeters cubed.

Parallelograms are easy if you understand the formula for triangles. The hardest part here, again, is to figure out what the height is. But we already know our clue is that right triangle symbol. The formula for area of a parallelogram is just base times height. We’re not going to multiply by one-half this time because a parallelogram is basically two of the same triangle put together. Since it’s so similar to a triangle, we’re not going to have an example for parallelogram. You should be able to handle plugging in the numbers into the formula.

The last polygon we’re going to look at in this lesson is the trapezoid. The formula for area of a trapezoid is one-half times the quantity base 1 plus base 2, times the height. In other words, we’re going to add the two bases together first, then divided by 2, then multiply it by the height. In a trapezoid, we have one pair of parallel sides. Those two parallel sides are our bases, base 1 and base 2. Again, we have the clue, for our height, it extends between the two bases at a perpendicular angle.

Let’s look at an example. Base 1 is 9 meters, base 2 is 7 meters. We know that because these two sides are parallel with each other. The height is 4 meters because it connects the two bases at a 90° angle. So I’m going to plug that into our formula here. One-half base 1 plus base 2, times the height which is 4. 9 plus 7 is 16. 16 divided by 2 is 8. 8 times 4 is 32. You’re going to keep the same units, meters, put the little 2 on top to show that it’s meters squared.

Sometimes you’ll have polygons with extra sides that you might not know what to do with. Well, if you can break that polygon up into smaller parts of shapes that you do know how to find the area for, you’ll be able to find the area of the whole thing. Here, we have a rectangle and two triangles. Find the area of the rectangle, the area of this triangle, and the area of this triangle. Add them all together and you have the area of the whole figure.

Let’s look at an example. If we add a line to this figure, we’ll break it up into two shapes – a rectangle and a trapezoid. The area of this rectangle is 8 times 4, which is 32, and the area of the trapezoid is the base plus the base, 12 plus 8 which is 20. 20 divided by 2 is 10, times the height -- 10 times 2 is 20. 32 plus 20 is 52. The units is feet squared.

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