Video: Areas of Composite Polygons

In this video, we will learn how to find areas of composite polygons including shapes made of squares, rectangles, triangles, and trapezoids.

17:22

Video Transcript

In this video, we will learn how to find the areas of composite polygons including shapes made of squares, rectangles, triangles, and trapezoids.

Firstly, we remember the definition of a polygon. It’s a closed shape with straight sides. So, for example, squares, rectangles, triangles, trapezoids, or trapeziums, and parallelograms are all examples of polygons. Whereas a circle isn’t as its sides are not all straight.

Composite polygons are simply two-dimensional shapes composed or made up of other polygons. And they can be composed in two ways. Firstly, we can join two or more polygons together. Or secondly, we could remove one or more polygons from another. To find the area of a composite polygon then, we need to consider either the sum or difference of the areas of the individual polygons it’s made up of. We, therefore, need to be comfortable calculating the area of some key polygons.

And we’ll review some of these formulas here. For example, for a square with a side length of 𝑙, we know that its area is equal to 𝑙 squared. And for a rectangle with a length 𝑙 and a width of 𝑤, the area is equal to the length multiplied by the width, 𝑙 times 𝑤. For a triangle, the area is a half the base multiplied by the perpendicular height. And for a parallelogram, it’s just the base multiplied by the height.

Finally, for a trapezoid or a trapezium, we take half the sum of the two parallel sides, which are often labeled as 𝑎 plus 𝑏, and multiply this by the perpendicular distance between them, which is called ℎ. So, the area is given by a half 𝑎 plus 𝑏 ℎ. Now that we’ve reviewed the key formulae for calculating the areas of some polygons, let’s consider some examples.

Calculate the area of the shape.

From the figure, we can see that we have a composite shape. And there are a number of different ways that we could approach this problem. One way would be to divide the shape horizontally into two rectangles. We know that the area of a rectangle is found by multiplying its length by its width. So, we just need to determine the dimensions of each rectangle, calculate their areas, and then add them together.

Rectangle one is actually the more challenging of the two. So, we’ll come back to it. Let’s consider rectangle two for now. We can see straightaway from the figure that it has a length of 12 centimeters and a width of two centimeters. The area of rectangle two then is, therefore, 12 multiplied by two, which is 24. And the units for this, which we’ll include at the end, are square centimeters.

For rectangle one, we need to determine the dimensions ourselves. Let’s consider the vertical distance, the width of the rectangle, first of all. Now, this will be the difference between the two sides of seven centimeters and two centimeters, the other two vertical sides in this composite shape. So, the width of rectangle one is seven minus two, which is five centimeters.

In the same way, the length or the horizontal side in rectangle one can be found as the difference of 12 centimeters and six centimeters, 12 minus six, which is equal to six. So, the area of rectangle one can be found by multiplying six by five, which gives 30. The total area, then, is the sum of these two values, 30 plus 24, which is 54. And the units for this area will be square centimeters.

Now, I did say that there were multiple approaches we could take for this problem. So, let’s consider another. Instead of dividing the shape horizontally to find two rectangles, we could instead divide it vertically to give a different two rectangles. We already determined the dimensions of each of these rectangles in our first method. So, we have that the area of rectangle one is six multiplied by seven, which is 42. And the area of rectangle two is six multiplied by two, which is 12. The total area, then, is the sum of these two values, giving 54 square centimeters once again.

Now, in fact, there is one final approach we could take, which is to consider this composite shape as the difference of two rectangles. The larger rectangle, outlined in green, has a length of 12 centimeters and a width of seven centimeters, giving an area of 84 square centimeters. The smaller rectangle, labeled rectangle two, has a length of six centimeters and a width of five centimeters, giving an area of 30 square centimeters. This time we find the total area of our composite shape by subtracting 84 minus 30, which is 54.

Using all three methods then, and you can choose whichever is most intuitive to you, we have found that the area of this composite shape is 54 square centimeters.

Let’s now consider a slightly more complicated example which involves polygons other than rectangles.

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.

In our next example, we’ll see how we can find the area of a composite polygon formed when one polygon is removed from another.

Below is a square which has had a rectangle measuring four centimeters by three centimeters removed. Find the area of the shaded part.

We have here a composite polygon. It’s one polygon, a square, and another polygon has been removed from it. We can, therefore, find this area as the difference of two areas, the area of the square minus the area of the rectangle. And we know how to calculate each of these areas. The area of a square is its side length squared. And the area of a rectangle is its length multiplied by its width.

From the diagram, we can see that the side length of the square, first of all, is seven centimeters. So, its area is seven squared. In the question, we’re given that the dimensions of the rectangle are four centimeters and three centimeters. So, its area is four multiplied by three. Seven squared is 49, and four multiplied by three is 12. So, the area of the shaded part is 49 minus 12, which is 37.

The units for this area will be square centimeters. And so, we have our answer to the problem. The area of the shaded part found by calculating the difference between the area of the square and the area of the rectangle is 37 square centimeters.

Let’s now consider another example in which the area we want to calculate is again the difference in the areas of two polygons.

Determine the area of the shown figure to the nearest tenth.

Let’s look carefully at the diagram. We can see, first of all, that we have a trapezoid or a trapezium. And from this trapezoid, a triangle has been removed to give the shaded region. The shaded area can, therefore, be calculated as the area of the trapezoid minus the area of the triangle.

We know the formulae for calculating the areas of each of these polygons, so we just need to determine their measurements from the diagram. Let’s consider the trapezoid first of all. 𝑎 and 𝑏 represent the two parallel sides of the trapezoid. We can see that one of the parallel sides is nine meters. And the other is the sum of the measurements of six meters, five meters, and six meters, which is 17 meters. We haven’t been given the perpendicular height of the trapezoid though. So, we’re going to need to find a way to calculate it.

In the triangle, we can see that the base is five meters. But, once again, we haven’t been given the perpendicular height. And I’m going to change the letters here so that we use capital 𝐻 to represent the height of the trapezoid and lowercase ℎ to represent the height of the triangle.

Let’s consider how we could calculate the height of this triangle. We can see from the diagram that it is an isosceles triangle because it has two equal sides each of nine meters. And so, we know that this perpendicular height divides the triangle into two identical right triangles. In each of these triangles, we know two side lengths. The length of nine meters, which is the hypotenuse as it’s opposite the right angle. And the length of 2.5 meters. That’s half the total base, five meters, of the original triangle.

As we know two of the side lengths and we wish to calculate the third, we can apply the Pythagorean theorem. Which tells us that in a right triangle, the square on the hypotenuse is equal to the sum of the squares of the two shorter sides, often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared. We can, therefore, form an equation. ℎ squared plus 2.5 squared is equal to nine squared.

We can then subtract 2.5 squared from each side. And evaluating nine squared minus 2.5 squared gives ℎ squared equals 74.75. ℎ is, therefore, the square root of 74.75. And as a decimal, this is a little over 8.6. But we’ll keep this value for ℎ in its exact form for now.

So, now, we know the height of the triangle, we can work out its area. It’s a half multiplied by five multiplied by the square root of 74.75. Remember, we’re back in the full isosceles triangle here not the smaller right triangle. For the area of the trapezoid, we have a half the sum of the parallel sides. That’s a half multiplied by nine plus 17. And the total height, capital 𝐻, will be the value we found for the height of the triangle plus the additional two meters.

So, we now have a calculation we can evaluate to find the shaded area. We can work each area out separately, if we wish, and then subtract, giving 116.780. The question asked for the area to the nearest tenth. So, rounding our answer and including the units, we have that the area of the shown figure to the nearest tenth is 116.8 square meters.

So, in this problem we combined our knowledge of the area of composite polygons with the Pythagorean theorem. Let’s now consider one final example.

Determine the area of the given figure.

Looking at the diagram, we can see that we have a composite polygon. So, we just need to carefully identify the individual polygons it’s made up of. Firstly, we have a trapezoid. We know that the two lines which appear to be horizontal are parallel because they each form a right angle with the dotted line which appears to be vertical. We then have two congruent shapes, which are in fact parallelograms. Although, they could each be divided into a right-angled triangle and another trapezoid.

We know the formulae for calculating the areas of each of these polygons. Let’s consider the trapezoid first of all. The two parallel sides 𝑎 and 𝑏 are 16 plus 16 meters, that’s 32 meters, and 46 meters. And the perpendicular height is 14 meters. So, substituting into the formula for the area of a trapezoid gives 546.

For each of the parallelograms, and it may help here to turn your head to the side slightly, we see that they have a base of 27 meters and a perpendicular height of 14 meters. We need to be careful here. It is the perpendicular height we need, which is 14 meters, not the slant height, which is 16 meters. Remember these two parallelograms are identical or congruent. So, we can find each area, 27 multiplied by 14, and then double it to give the total contribution, which is 756.

The total area then is 546 for the area of the trapezoid plus 756 for the area of the two parallelograms, which is 1302. And the units for this area will be square meters.

Don’t worry if it sometimes takes you a little bit of time to identify the individual polygons that make up a composite polygon. It’s worth spending the time to do this and find the most efficient approach. You may find it helpful to tilt the diagram and look at it in different orientations.

Let’s now review some of the key points from this video. Firstly, a polygon is a straight-sided shape. A composite polygon can be formed in two ways, either by joining two or more polygons together or by removing one or more polygons from another. The area of a composite polygon can, therefore, be found by considering either the sum or the difference of the areas of the individual polygons that it is composed from. In order to work with composite polygons, we must, therefore, be familiar with the formulae for calculating the areas of individual polygons, including squares, rectangles, triangles, trapezoids, and parallelograms.

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