Video: Finding the Area of a Composite Figure Involving Triangles and Rectangles

Determine the area of the figure.

04:07

Video Transcript

Determine the area of the given figure.

From the diagram, we can see that we have a composite polygon; it’s made up of other polygons. There are a couple of different approaches that we could take. One approach would be to introduce a second vertical dividing line, which would separate this polygon into two congruent, that is, identical, triangles and a rectangle.

We know the formulae for calculating each of these areas, so it’s just a case of determining the dimensions of each polygon. For the rectangle, we can see from the figure that its length is 12.6 centimeters and its width is 12 centimeters. So, its area will be 12.6 multiplied by 12.

For the two triangles, we can see that they have a base of 12 centimeters. But what about the perpendicular height? Well, we can work this out using the measurements of 18 centimeters and 12.6 centimeters on our diagram. Remember these two triangles are congruent, that is, identical. We know this because of the little blue lines indicating that two of their sides are the same length.

This means that the total length of the figure, 18 centimeters, will be made up of the length of the rectangle, 12.6 centimeters, plus the perpendicular height of these triangles on either side. We can, therefore, form an equation, ℎ plus ℎ plus 12.6 is equal to 18. To solve our equation, we can subtract 12.6 from each side, giving two ℎ equals 5.4, and then divide each side of the equation by two to give ℎ equals 2.7. So, we know the perpendicular height of our triangles.

The area of each triangle is, therefore, 12 multiplied by 2.7 over two. And remember there are two of them. We now have a calculation we can use to work out the area of this figure. Firstly, we can cancel a factor of two in the numerator and denominator of our triangle calculation. We can then work out 12 multiplied by 2.7 and 12.6 multiplied by 12 either using a calculator if we have one or using a method such as the grid method. And they give 32.4 and 151.2.

Adding these values together gives 183.6. And the units for this area will be square centimeters. So, that’s one method we could use. But it’s always good to consider multiple approaches where possible.

Another approach we could take is to divide our composite figure horizontally. And when we do, we see that we have two congruent, that is, identical, trapezoids. We know that the area of a trapezoid is given by a half multiplied by 𝑎 plus 𝑏 multiplied by ℎ, where 𝑎 and 𝑏 represent the parallel sides and ℎ is the perpendicular distance between them.

From the diagram, we can identify that the parallel sides of our two trapezoids are 18 centimeters and 12.6 centimeters. The perpendicular height of the trapezoids will be half the total height of the figure, which was 12 centimeters. So, the height is six centimeters. Substituting these values then, and we have that the area of each trapezoid is a half 12.6 plus 18 multiplied by six. But remember, there are two identical trapezoids.

So, we have our formula for calculating the area of this figure. And we can begin by canceling a factor of two from the numerator and denominator. 12.6 plus 18 is 30.6. And we can then multiply this by six in a variety of different ways. I’ve chosen to multiply 30 by six and 0.6 by six and then add these values together, which, of course, gives the same answer as our previous method of 183.6 square centimeters. We’ve seen in this question that there’s usually more than one approach we can take to finding the area of a composite polygon.

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