Video Transcript
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.
