Using 3.14 as an estimate for 𝜋, calculate the area of the given figure.
Remembering that the area is the space within a figure, the first thing we need to do here is identify the different shapes that make up this composite figure. The shape on the left is a semicircle, and this shape on the right is a trapezoid. We know this to be true because it’s a quadrilateral and it has a pair of parallel sides.
In order to find the area of this figure, we’ll need to remember two important formulas. Firstly, the area of a circle is equal to 𝜋 times the radius squared. And we remember that it’s just the radius that’s squared, and it doesn’t include the 𝜋. The second formula is that the area of a trapezoid is equal to half times the height times 𝑏 sub one plus 𝑏 sub two, where 𝑏 sub one and 𝑏 sub two are the lengths of the parallel sides.
So let’s start by finding the area of the semicircle. To do this, we’ll take the formula for the area of a circle and half it. And this will be 𝜋𝑟 squared over two. The radius of a circle is the distance from the center to the outside. And we can see that this value of nine millimeters is given on the diagram. We can plug this into the formula along with the fact that we’re told to use 3.14 for 𝜋 to give us 3.14 times nine squared over two.
We can simplify this calculation and evaluate nine squared as 81 to give us 1.57 times 81. We can work this out using a non-calculator method by calculating 157 multiplied by 81 and then remembering that our answer will have two decimal places. So we find the area of the semicircle to be 127.17 square millimeters. Next, we find the area of the trapezoid using the formula. We can plug in the values of the height as 13 millimeters. One of the parallel sides is seven millimeters, but what about the other side?
Well, we know that we’ll have two radii here, so that means that the base will be nine plus nine, which is 18. So our calculation is a half times 13 multiplied by seven plus 18. We can evaluate this using any method. We could, for example, take 13 and multiply it by 25 and then find half of it. Or we could find a half of 13 and then multiply it by 25. Either way, we can use a non-calculator method to find the answer of 162.5. And the units here will still be our area units of square millimeters.
Finally, to find the total area of this given figure, we add together the area of the semicircle and the area of the trapezoid to give us the answer for the area of the given figure of 289.67 square millimeters.