Understanding Shaded Areas

Have you ever pondered how to calculate the area of a shaded region? Will you lean towards calculus or geometry for the solution?

While I usually don't focus on politics, I found myself listening to the US presidential debate while composing this puzzle—quite entertaining! Now, let's get back to the question at hand: will you tackle this geometry challenge using calculus or a geometric approach?

I encourage you to pause here, grab some pen and paper, and attempt to solve it. Once you're ready, continue for the solution!

Solution

I prefer straightforward and organized methods, so I'll demonstrate the geometric solution. We can simplify the shaded area by dividing it into three components.

The shaded region consists of two right-angled triangles flanking a sector in the center. First, let's calculate the areas of the two triangles.

To find the height, we can apply the Pythagorean theorem, where the hypotenuse corresponds to the radius length, which is 1 unit.

Next, we need to determine the area of sector B. This requires us to find the angle it subtends within the circle. By subtracting the angles of the two triangles from 180 degrees, we can find this angle.

Using trigonometric relations, we arrive at

This gives us an angle of approximately 74.43 degrees for the sector. To compute the area of the sector, we can use the following formula:

With the assistance of Google’s calculator, we can then find

And there you have it! Have you discovered the calculus method yet?

Photo by Guido Coppa on Unsplash

Isn’t it fascinating?

What was your thought process while tackling this challenge? Please share your insights in the comments; I'm eager to hear from you!

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This video explains how to calculate the area of shaded regions in concentric circles.

This video covers finding areas of shaded regions involving circles, rectangles, triangles, and squares in geometry.