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.
