Video: Areas of Composite Figures

In this video, we will learn how to find the area of a composite figure consisting of two or more shapes including polygons, circles, half circles, and quarter circles.

13:42

Video Transcript

In this video, we will learn how to find the area of composite figures consisting of two or more shapes, including polygons, circles, half circles, and quarter circles.

A composite figure is simply a figure or a two-dimensional shape that’s constructed from two or more geometric figures. For example, if we had a decorative window and we wanted to find the area of this shape, we could notice that it’s divided into a semicircle and a rectangle. And therefore, the area of this composite figure would be the sum of these areas.

We might often find that there’s more than one way to calculate the area of a composite figure. Let’s say we wanted to calculate the area of this composite shape. One way we could calculate this is by calculating the area of the two rectangles internally. Or we could also calculate the area by finding the area of the larger rectangle and then subtract the area of the smaller orange rectangle that would give us the area of the blue shape. So when we’re finding the area of a composite figure, sometimes we might need to add the areas, sometimes subtract, and sometimes even a mixture of both.

As we’ll be looking at the areas of polygons and other shapes, let’s remind ourselves of some of the key formulas. The area of a rectangle is the length multiplied by the width. The area of a triangle is half times the base times the perpendicular height. The area of a circle is equal to 𝜋 times the radius squared. And we can remember that it’s just the radius that’s squared, and it doesn’t include the 𝜋.

Knowing this formula will also allow us to work out the area of parts of a circle. For example, the area of a semicircle is equal to 𝜋𝑟 squared over two. And if we wanted to find, for example, the area of a quarter circle, we would have 𝜋𝑟 squared divided by four. And finally, the area of a trapezoid is equal to 𝑎 plus 𝑏 over two multiplied by ℎ, where 𝑎 and 𝑏 are the lengths of the two parallel sides and ℎ is the perpendicular height.

We’ll now look at some questions where we find the area of a composite figure.

Using 3.14 as an estimate for 𝜋, find the area of this shape.

We notice that this composite figure is made up of a triangle and a quarter circle. So to find the area of the shape, we need to calculate the area of the triangle and the area of the quarter circle and add them together. Beginning with the triangle, we can recall that the area of a triangle is equal to half times the base times the height. In the triangle, we can see that we have a length of 28.5, which we can take as the base. For the height then, as we can see that this is part of the quarter circle with a radius of 14 centimeters, then this means that the height of our triangle will also be 14 centimeters. So, we’ll be calculating a half times 28.5 times 14. We can simplify this calculation to 28.5 multiplied by seven, which evaluates as 199.5. And as we’re working with an area, we’ll have the square units of square centimeters.

Next, we can find the area of the quarter circle. And we can recall that the area of a circle is 𝜋𝑟 squared. So therefore, the area of a quarter circle would be a quarter of this, which is 𝜋𝑟 squared over four. Substituting in the value of 14 for the radius, we’ll then have 𝜋 times 14 squared over four. As 14 squared is 14 times 14, we’ll have 196𝜋 over four. This simplifies to 49𝜋. And as we were told to use 3.14 as an estimate for 𝜋, then we can evaluate 49 times 3.14 as 153.86. And our units here will still be in square centimeters. We can now use these two areas to find the total area. So we add the area of our triangle, 199.5, to the area of our quarter circle, which was 153.86. And so our final answer is 353.36 square centimeters.

In the next two examples, we’ll see how you must subtract the areas within a composite figure to find the remaining area.

Determine, to the nearest tenth, the area of the shaded region.

We begin by noticing that this circle is inscribed within a square. So to find the area of the shaded region, we would begin by finding the area of the square and then subtract the area of the circle within it. To find the area of a square, we multiply the length by the length. But what would this length be? Well, as the circle sits exactly within the square, this means that the lengths of all the sides of the square would be the same as the diameter of the circle, which is 19.7 yards. And so our area is 19.7 times 19.7, 388.09. And as it’s an area, our units will be squared, so we’ll have square yards. Then, to find the area of the circle, we recall that this is equal to 𝜋 times the radius squared. We’ll need to be careful when we’re plugging in the values here as the radius is not 19.7 because that’s the diameter.

To find the radius, we half the diameter, so we’ll be calculating 𝜋 times 9.85 squared. We can then put that directly into our calculator, being careful just to square the 9.85 and not the 𝜋 as well, giving us a value of 304.805173 and so on square yards. We won’t round this decimal value yet until we reach the final stage of the question. Putting these together then to find the area of the shaded region, we’ve got the area of our square and the area of our circle. So we have 388.09 subtract 304.805 and so on, which gives us 83.284826 and so on square yards. And since we’re asked to round it to the nearest tenth, that means we check our second decimal digit and see if it’s five or more. And as it is, then our answer rounds up to 83.3 square yards for the area of the shaded region.

Using 3.14 as an approximation for 𝜋, find the area of the shaded shape.

If we look at the diagram, we can see from the shading that this shape in orange is excluded from the area. This orange shape is a semicircle. As this shaded figure or shape is composed of two geometric figures, then we can call this a composite figure. We can calculate the area of this shaded region by finding the area of the large rectangle and then subtracting the area of the semicircle. Let’s begin by calculating the area of this rectangle. And as we find the area of a rectangle by multiplying the length by the width, we’d have 52 times 23. We can evaluate this without a calculator to give us the value 1196. And the units here as it’s an area will be squared centimeters.

Now let’s focus on the area of the semicircle. We can use the fact that the area of a circle is equal to 𝜋𝑟 squared to establish that the area of a semicircle would be half of that. In other words, it’s 𝜋𝑟 squared divided by two. Before we can use this formula, however, we need to work out the value of the radius of this semicircle. The length of our rectangle is 52 centimeters. And therefore, we can see that the diameter of this circle can be found by subtracting the other two lengths of 15 from 52, which gives us the diameter of 22 centimeters. And so the length of our radius will be half of that. Half of 22 will give us 11 centimeters. Plugging this value into our formula for the area of a semicircle gives us 𝜋 times 11 squared over two, which is 𝜋 times 121 over two.

In order to evaluate this, we were told to use 3.14 as an approximation for 𝜋. We could put this directly into our calculator if we wished. But if we weren’t using a calculator, then we could simplify the calculation to 1.57 multiplied by 121 and then use any method of multiplication to work this out. Here, I’ve used the grid or area method to find the value of 189.97 square centimeters for the area of the semicircle. Now that we’ve found the two important pieces of information, the area of the rectangle and the area of the semicircle, we can work out the area of the shaded shape.

Remembering that as this shaded shape does not include the area of the semicircle, this means we must subtract it from the area of the rectangle. Therefore, we’ll have the calculation 1196 subtract 189.97, which gives us 1006.03 square centimeters. And that is our final answer for the area of the shaded shape.

In these questions, so far, we have seen how two shapes can make up a composite figure. In the final question, we’ll see an example of a more complex composite figure, which is made up of four shapes.

Determine, to the nearest tenth, the area of the given figure.

We can see from the diagram that this figure is a composite figure. That simply means that it’s made up of two or more geometric figures. Let’s see if we can work out what these different shapes are. Well, the three shapes at the top are semicircles. Notice that these three semicircles are all the same size or congruent. We can say this because the marking on the radii shows that these are all the same lengths in the three semicircles. Looking at the shape on the lower part of this figure, we could say that it’s composed of a rectangle and two triangles. However, there is a much easier shape. And that is that we can identify that this is a trapezoid.

And so to find the area of the entire figure, we’ll need to find the areas of the three semicircles and the area of the trapezoid and add them together. Let’s begin by finding the area of the semicircles. We can recall that the area of a circle is equal to 𝜋 times the radius squared. And so therefore, the area of a semicircle would be half of that, 𝜋𝑟 squared over two. Before we use this formula, we need to find out the value of the radius as this value of 10 inches refers to the diameter. As the radius is half of the diameter, this means that our radius is five. And so we have the calculation 𝜋 times five squared over two. We can write this as 25𝜋 over two. As this is an area, our units will be squared, so we’ll have square inches.

We can keep our answer in terms of 𝜋 as we’ll use it in the final calculation. But if we did change it into a decimal, then we wouldn’t round that value yet. In a moment, we’ll be able to use this value for the area of a semicircle to find the area of three semicircles. But let’s move on to finding the area of the trapezoid. The area of a trapezoid is calculated by 𝑎 plus 𝑏 over two times ℎ, where 𝑎 and 𝑏 are the two parallel sides and ℎ is the perpendicular height. Filling in the values then for our trapezoid, we have a length of 20. And the other parallel length is formed of three lots of 10 inches.

We can say this because we know that we had three congruent semicircles that all had a diameter of 10 inches. And we also have a perpendicular height of 20 inches. We then have a calculation of 50 over two multiplied by 20, which simplifies to give us the answer of 500 square inches. We’ve now found enough information to help us calculate the area of the entire figure. As we calculated that the area of one of these semicircles is 25𝜋 over two, then as we established that our semicircles are congruent. To find the area of three semicircles, we would multiply the area of one semicircle by three. And to this we add the area of our trapezoid, which was 500 square inches. We can use our calculator to evaluate this as 617.8097245 and so on square inches.

But to complete our answer, we must round to the nearest tenth. And as our second decimal digit is not five or more, then our answer stays as 617.8 square inches. And so we found the area of this figure by adding together the individual areas of each of the shapes within it.

We can now look at some of the important things that we learned in this video. We learned firstly that a composite figure is made up of two or more geometric figures. To find the area of a composite figure, we separate it into simpler shapes whose area can be found. And finally, when we’re finding the area of a composite figure, sometimes we need to add the individual areas, and sometimes we need to subtract them.

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