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Lesson 153: Area of Overlapping Figures
We've learned how to find the area of various two dimensional figures. In this lesson we'll learn how to find the area of a shaded region, by the subtracting the area of an inner shape from the area of an outer shape.
This lesson is quite tricky, and many students have trouble with this topic. The problems in this topic really test your total understanding of math. Very often you need to think creatively, and do many steps to get your final answer. With that said, in this lesson, I will go through many of the steps quickly, because we have already covered the necessary material. Please review the corresponding lessons if necessary.
Let's take a look at three typical types of problems for this topic. The first type of problem is as follows: Two concentric circles have radii (plural of radius) of 7 and 10. Find the area of the shaded region between the circles as shown in the diagram. Note that concentric means that one circle is within the other, and they share the same midpoint.
To solve problems of this nature, we always first find the area of the outer (larger) region, and then subtract the area of the inner (smaller region). We know the area of a circle is found by πr2. That means that the outer circle has area of 100π. The inner circle has area 49π. The area of the shaded region is 100π - 49π, which is 51π.
Remember that π is just a number, a particular constant. That means we can subtract in the way that we did. The final answer is left in terms of π, and is simply 51π. Officially, you should write 51π units2. We don't know the units in this problem, so we have to leave it like that. If we were told that the radii were in centimeters, our answer would be 51π cm2.
Here is another typical shaded area problem. Find the area of the shaded area shown in the diagram at right. This is quite tricky. We have a circle, which means we should start thinking about finding its radius. How can we do that? Notice that the diagonal of the rectangle will cross directly through the center of the circle. That means that the diagonal of the rectangle is actually the diameter of the circle. Once we know the diameter, of course it is easy to find the radius.
How do we find the diagonal of the rectangle? If we look carefully, we can see that drawing a diagonal actually forms a right triangle, and the diagonal is the hypotenuse. That means we can use the Pythagorean Theorem to find it.
I will skip that step, because we've done that in previous lessons. The legs are 6 and 8. Using the Pythagorean Theorem, we find the the hypotenuse, and therefore the diagonal of the rectangle, and therefore the diameter of the circle is 10. That means that the radius is 5.
Continuing, we find that the area of the circle is 25π. The area of the rectangle is found by multiplying 6 times 8, giving us 48. Our final answer is 25π - 48 units2, which is how we must leave the answer.
There are many other possible problems of this type, but they all follow this general pattern. Find the area of the outer region, and subtract the area of the inner region. Utilize all of your math knowledge to help you, and look at the problem as creatively as you need to. These problems take lots of practice. Do as many as you can, and write to me if you need help.