Finding the Area of Parallelograms and Trapezoids

The mayor of Polygon, Pennsylvania is launching a "Revitalize the City" campaign to engage people in projects that make their city a better place to live. Today, she's meeting with a citizen who wants to bring some color and flair to some of the city's many dumpsters! It's an opportunity to turn a trash heap into a cultural epicenter. All it will take is a little paint, hard work, and, well two methods for using triangles to find the area of trapezoids. The sides of all of the city's dumpsters are shaped like trapezoids. A trapezoid is a quadrilateral where at least two of the four sides are parallel. This dumpster has parallel sides of 8 feet and 12 feet. It also has a height of 5 feet. In order to determine how much paint will be needed to give this dumpster a makeover, we're going to need to find the area. Let's try a strategy you may have used before. Imagine decomposing this trapezoid into smaller shapes which will help us find the area. Hmm, this isn't very helpful. Neither is this. But what if we cut along the diagonal? Now to get the trapezoid's total area, we can just add the area of these triangles together. We'll call the bottom triangle, triangle 1 and the top triangle, triangle 2. What information do we have about triangle 1? Well we know that this side has a length of 12 feet. Let's call that our base. We also know that the perpendicular distance from the base to the top of the triangle is 5 feet. That's our height. Remember the base and height of a triangle are always perpendicular to each other. That is indicated by this right angle sign, here. Now how can we use that information to find the area of triangle 1? Remember that the area of any triangle can be found by multiplying one-half the base, 'b', times the height, 'h'. Substituting these values into our formula triangle 1 has an area of 30 square feet. Now let's look at triangle 2. The area of this triangle is also going to be one-half the base times the height. So what's the base? Well we only know one side length, so let's use that for our base. Now what should we use for our height? Notice that because the bases we used are parallel, the heights of both triangles are the same. Multiplying one half our base and height gives us an area of 20 square feet for triangle 2. Adding the area of our two triangles together gives us a total area for the trapezoid of 50 square feet. Now to give this dumpster a splash of color! For our next example, here's another design by our local citizen. This one is painted on a rectangular canvas, and she wants to use glue to attach it to a dumpster. So we're going to need to use triangles again to figure out the area of a trapezoid. This canvas is composed of three shapes: the trapezoid in the middle and two triangles on the sides which we'll call triangle 1 and triangle 2. Our plan is to first find the area of the enclosing rectangle then subtract the areas of the two triangles leaving us with the area of the trapezoid. We can express this process with the formula: total area equals area of the rectangle, minus the sum of triangle 1 and triangle 2. Let's start by finding the area of the rectangle. Multiplying 12 feet times 5 feet gives us an area of 60 square feet. Let's plug that value into our formula before we move on. Now to figure out the area of triangle 1. What values should we use for the base and height? The base and height are perpendicular to each other so the area of this triangle will be one-half 3 feet times 5 feet. That gives us 7.5 square feet which we can substitute into our equation. Now, we can turn our attention to the final piece of the puzzle: triangle 2. Triangle 2 has a base of 1 foot and height of 5 feet. That makes its area 2.5 square feet. Substituting that into our equation. Then simplifying we get the area of the trapezoid, 50 square feet. Now, while the dupster artist puts the finishing touches on her work let's review our methods. We looked at two strategies using triangles to find the area of trapezoids. For the first strategy, we can decompose a trapezoid into triangles by drawing a diagonal from opposite corners. Then, using the area formula one-half base times height we find the area of both triangles and add them together. For our second strategy we compose a rectangle around the trapezoid, creating two new triangles. Then, we find the total area of the enclosing rectangle and subtract the areas of the two new triangles. Which method you use depends on the information you are given and of course, which approach you prefer. It's time for the grand dumpster unveiling! Wow, what a difference a little paint can make. Now the citizens of Polygon are lined up for hours just to throw away their trash.