Trying to Explain Math Problems

The last question on the 2022 British GCSE maths exam
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I saw this video about a geometry problem the other day and within a couple of seconds I had figured out how to solve it. Okay, good for me, but then I start watching this dude's explanation and I wonder what the devil is he doing? The solution is so simple, why is he making it so hard? So I sat down to do the calculations for my method and I had to do it two or three times before I got the right answer. You've got to be paying attention when you are working problems like this and being as I am retired (I.e. a professional slacker) it takes a bit to get the ol' brain running on all cylinders. 

So now that I have verified that my technique works, can I explain it to anybody else? The technique in the video is pretty good, just use dry-erase markers on a whiteboard and record your explanation with a video camera. I've got a white board and my smart phone has a video camera. Probably have to order some dry-erase markers and buy or make a tripod. But then you have to record your explanation, and since the first take is going to be garbage, you're going to have to do it again, possibly several times, and you are going to have to watch each of them to gauge whether they are any good or not. Bah, sounds like a giant time suck. I'm a keyboardist, probably should stick to what I know.

So now I'm looking for math symbols I can use in this blog and I found the code for Π (pi) and for √ (the radical for square roots). Funny thing is you put the html code in and Blogger turns it into a character.

Most math equations like to represent division with a horizontal line with one number above and one number below. That's great if you have an editor that supports it, but it takes extra special fiddling to do that using Blogger's editor. Computer code uses a / (slash) to represent division. It isn't as pretty and often requires parentheses, but since I'm getting tired of mucking around here, it's good enough. The radical symbol doesn't automatically carry over all the following digits, so I'm using parentheses here as well.

The last problem is what to call the slice of a circle that is cut off by a chord. There ought to be term for that. Wikipedia calls it a circular segment. That's still too long for me. I'm going to call it a sword cause it kind of reminds me of the scimitars from ancient Arabia.

Here's the original problem:

Find the area of the shaded portion

Here's the same three circles with a couple of equilateral triangles:

Geometry Problem

You can see that each of the two shaded areas are a pie shape with two swords cut out of the sides. The area of the sword is the difference between the area of a pie shape and the area of an equilateral triangle. So all we have to do is compute the area of a pie shape and the triangle and then do a little addition and subtraction. Here's the math. r, the radius of the circle, is 4.

Area of pie shape = Πr²/6

Height of equilateral triangle = √(r² - (r/2)²)

 = √(r² - r²/4)

 = √(4r²/4 - r²/4)

 = √((r²/4)4 - (r²/4)1)

 = √((r²/4)(4 - 1))

 = √(r²/4) √(4 - 1)

 =    (r/2)  √(4 - 1)

 =    (r/2)  √3

Area of equilateral triangle = 1/2 x base x height

= r * (r/2) √3 / 2

= r² * √3 / 4

Area of sword (space between the triangle and the circle)

= Area of pie shape - Area of equilateral triangle

= Πr²/6 - r²√3 /4

= r²(Π/6 - √3 /4)

Area of one shaded area

= Area of pie shape - 2(Area of sword)

= r²Π/6 - 2r²(Π/6 - √3 /4)

= r²Π/6 - r²(Π/3 - √3 /2)

= r²(Π/6 - Π/3 + √3 /2)

= r²(√3 /2 - Π/6)

Area of total shaded area

= 2(Area of one shaded area)

= 2r²(√3 /2 - Π/6)

= r²(√3 - Π/3)