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.
