What Is The Area? HARD Geometry Problem

People around the world have asked me this and similar problems. Learn how to solve using only geometry (and a little bit of algebra).

There are many other ways to solve the problem too

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[Its__Good]

The relief when you give up quickly, then find out that the solution involves a mathematic rule you didn't know.

[rational-ecrk]

"Trigonometry or Calculus not allowed"

Me: Can't solve using Calculus or Trigonometry either.

[Ozymandias]

“Find the answer without using any of the appropriate methods to do so”

[ArtumTsumia]

The 'fun' part about a problem like this is so long as you know how to calculate area of a square and circle, you can slowly break the area up into it's sections. So you can solve for the square, then the quarter circle, then subtract to get the 'outside' of the circle. Then you can take the area of half the square (the triangle) and proceed to continue to break things down. It's more roundabout than knowing some of the angle formulas going in, but they can be derived from the core area formulas.

[evescent]

I was like, integrals are a thing..
And then, "no calculus"
.
☺

[joshualiley]

"No trigonometry allowed"
*Procedes to use trigonometry*

[nberz]

The worst thing, when you know all this formulas and understand what is going on, but you didn't solve it

[Mariorox]

I probably spent longer doing the trig. version than I would have if I had done it as he did. Knowing that arch segment actually helped me, since I could look at a shape and go "Oh, that's a triangle with two of those arch shapes taken away"

I also love how the arcs perfectly trisect themselves.

[Wagon_Lord]

When you get your equilateral triangle, you can see quite clearly that the area of the square is [Eq. Triangle+ two 30 degree circular arcs + region a]. This lets you quickly find an expression for a, then you can find an expression for b by using simple algebraic manipulation, and again find the area of c. Nice to see I got the same answers at least.

[light__yagami]

Can't find the third equation but after watching the approach it's good

Al-Kashi did not discover the Law of Cosines, he just expressed one of Euclid's theorem in terms of cosines. If anything, it's Euclid's Law of Cosines.

[mx]

It is nice to generalize area a, b, & c. A slight different approach is just use variable a & b: 2a+b=square - quarter of the circle; a + b = 1/12 of the circle - (1/6 of the circle - the equilateral triangle). This will reduce the equations to two variables and two equation.

[gifropan]

In working out the area of the equilateral triangle haven't we used trigonometry? Because the square root of 3 divided by two is the cosine of 30 degrees.


I enjoy your videos immensely.

[mr.goldfarmer]

i remeber this question coming up in a maths text book 25 years ago. Stumpped me at the time. Incredible solution!

[technodrome]

I look forward to solving this set of questions. I love the circle and square area problems.

[mubtasimnawarmubtasimnawar]

i was thinking so much of sectors that I forgot that joining the radius(r) together forms an equilateral triangle. Thank you soo much for this video and explanation!!!!

[bgetther]

The hardest part of this is to figured that out yourself in limited time.

[magnetoninja]

A quick way is to look at it as a square and 4 "arc circles". Then by symmetry and 30-60-90 triangles, you notice that the arc is going to have a central angle of 30 degrees, you calculate the "arc circle" by finding the sector minus the triangle (1/2absinC) and then find the square using LoC. Basic techniques and quick :)

[sujalarora]

Though I was not able to solve in a general way, but solved using calculus.
When i did it correctly, it gave me much pleasure.
I think it's a great feeling for a guy in 11th class

[Trixex]

Had a similar problem in my a level further maths book, I used this method and was surprised with how hard the question was. Then it was when I realized that there was an eazier way to do it