Can You Find the Area? | Geometry Challenge with Trigonometry

[00:00:00]Have [music] you ever looked at a geometry problem and thought, "There's no way I can find that shaded area?"

[00:00:07]Well, let's [music] prove that's not true.

[00:00:10]Today's challenge is simple. Find [music] the total area of the two green semicircles.

[00:00:16]Pause the video if [music] you'd like to solve it first.

[00:00:25]Ready?

[00:00:26]Let's begin.

[00:00:27]The first [music] thing to notice is that these diagonals create two equilateral triangles.

[00:00:33]Each triangle [music] has a side length of eight. That means every interior angle is 60°.

[00:00:41]Now, look at one of the green semicircles. [music] Many people immediately assume the radius is four, but that's not true. [music] We actually have to calculate it.

[00:00:51]Notice that the center of the semicircle is exactly at the midpoint of the side.

[00:00:56]Now, draw [music] a perpendicular line from the center of the semicircle to the slanted side of the triangle. [music] Why?

[00:01:05]Because the radius of a circle is always perpendicular to its tangent.

[00:01:10]>> [music] >> That perpendicular segment is exactly the radius we're looking for.

[00:01:15]Now, we have a right [music] triangle.

[00:01:17]The side adjacent to the 60° angle is four because the center [music] lies exactly at the midpoint of the side.

[00:01:24]Now, let's use trigonometry.

[00:01:27]We know that sine 60° equals opposite over hypotenuse.

[00:01:33]In our triangle, [music] the opposite side is the radius, R, and the hypotenuse is four.

[00:01:39]So, sine of 60° equals R over four.

[00:01:44]Multiply both [music] sides by four.

[00:01:47]R equals four times sine of 60°.

[00:01:51]Since sine [music] of 60° equals square root of three over two, the radius becomes [music] r = 2 square root of 3.

[00:02:02]Now, the hard [music] part is over.

[00:02:04]The area of one semicircle is pi r squared over [music] 2. Substituting the radius, pi 2 square root of 3 squared [music] over 2 = 12 pi over 2, which is 6 pi.

[00:02:18]Since there are two identical [music] semicircles, we simply multiply by two.

[00:02:23]So, the total shaded area is 12 pi square [music] units.

[00:02:28]And that's it.

[00:02:29]Notice something interesting. We never needed the rectangle's diagonal. The only things we [music] really needed were the properties of an equilateral triangle, a little trigonometry, [music] and the area formula for a circle.

[00:02:43]That's the beauty of geometry.

[00:02:45]The information that looks important isn't always the information you actually need.

[00:02:51]If you enjoyed this geometry challenge, subscribe to Math Algo. We'll [music] keep turning difficult-looking problems into simple step-by-step solutions.

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