Video: Finding the Area of a Composite Shape Including Semicircles and a Trapezoid

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


Video Transcript

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.

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