What Is The Area?

This is adapted from a problem given to 14-16 year old students. Watch the video to learn how to solve it!

Source (in Spanish)

Send me suggestions by email (address in video). I consider all ideas though can't always reply!

Why are some comments before the video is published? Get early access and support the channel on Patreon

If you buy from the links below I may receive a commission for sales. This has no effect on the price for you.

Show your support! Get a mug, a t-shirt, and more at Teespring, the official site for Mind Your Decisions merchandise:

My Books
Mind Your Decisions: Five Book Compilation
A collection of 5 books:
"The Joy of Game Theory" rated 4.1/5 stars on 44 reviews
"The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias" rated 3.5/5 stars on 4 reviews
"40 Paradoxes in Logic, Probability, and Game Theory" rated 4.4/5 stars on 13 reviews
"The Best Mental Math Tricks" rated 4.7/5 stars on 8 reviews
"Multiply Numbers By Drawing Lines" rated 4.3/5 stars on 6 reviews

Mind Your Puzzles: Collection Of Volumes 1 To 3
A collection of 3 books:
"Math Puzzles Volume 1" rated 4.4/5 stars on 13 reviews
"Math Puzzles Volume 2" rated 4.5/5 stars on 6 reviews
"Math Puzzles Volume 3" rated 4.1/5 stars on 7 reviews

Connect with me

2017 Shorty Awards Nominee. Mind Your Decisions was nominated in the STEM category (Science, Technology, Engineering, and Math) along with eventual winner Bill Nye; finalists Adam Savage, Dr. Sandra Lee, Simone Giertz, Tim Peake, Unbox Therapy; and other nominees Elon Musk, Gizmoslip, Hope Jahren, Life Noggin, and Nerdwriter.

[sporksto]

That line can't be 18. This question is incorrect on purpose for certain.

[OptimusPhillip]

Pain is realizing you spent a half hour applying quadratics and geometry to a problem that is geometrically impossible.

[andyiswonderful]

You can just look at the diagram, and tell that those are impossible lengths.

[TatanAgustin]

The line were the 10 is, is the longest possible line perpendicular to the large circle to the smallest one, therefore the lengths are impossible.

[alfewenxiao]

The first thing I thought of when I saw the thumbnail:
That can't be right

[mathmelody]

The most important issue in circles.
[ KEY POINT 1 ] Radius
[ KEY POINT 2 ] Pythagorean theorem
[ KEY POINT 3 ] Similarity
Remember!!!

[markphc]

Well, I thought that the labelled lengths were screwy immediately, so I didn’t know what was going on

[oldguydoesstuff]

For a change, I saw the "trick" part of the problem almost instantly. The way the original problem is set up, the distance on the right side of the two circles must be greater than the distance above (or below) the circles. As given, the construction is impossible. But you could make the construction possible again by setting the TOTAL distance of the above and below portions to equal 18. That would make each portion 9, which is less than 10, and the problem would be solveable.

[jackhiggins]

Would’ve been nice to see you derive, or at least show, the allowable values of a and b. I found them to be a < b < 2a

[studies]

Woww...
The question you first solved is even not solved by graduate and post graduate students..The way these questions solve we could not even imagine
Because we dont see these type of questions before ..
Thanks for providing us such knowledge without any expense....

[et]

Give me a break, it should take exactly 1 second to know the question is wrong

[advaitpetiwale]

Getting recommended a Mind Your Decisions video from last year where Presh still used the term "Pythagorean Theorem" is a blessing.

[iamyoda]

You should've used *intersecting chords theorem.* It would be easier.

[kcpal]

This is the actual method to prove that the question is wrong.. thanks dear Paresh..sir..

[cakatikaggarwal]

Well another thing that came to mind directly when I saw you first problem was that
When circles are arranged like this .. the distance shall be maximum between the two circles at the horizontal gap line you have placed on the right size of the diagram between the two circles. .. so from starting it self I was skeptical to have a number 10 there and 18 on the vertical gap lines
Good one 😊👍👍

[antonsmirnov]

I think you can you solve the problem using another method, which may be even simpler. A circle geometric theorem states that the product of intercepting chords are equal. Therefore if we denote R as the larger radius, R(R-18)= (R-10)^2. R^2-18R=R^2-20R+100. 2R=100, R=50.

[onthespotstem]

No one:
Mindyourdecisions: a middle schooler could solve this in under a minute! can YOU solve this problem?

[surajsapkal]

Yeah i was skeptical in the beging but any way applied the formula

[arikwolf]

My first though was this can't be possible. Unless the two vertical distances sum to 18 and are actually 9 each. Just look at the drawing, the horizontal distance of 10 must be the max. Proving that would make for an interesting problem, but that is not what was asked.

[sadaa_aabhaari]

My method is dffferent.
• Perpendicular chord theorem leading to the value of the bigger radius
• then substituting the value to find diameter of the smaller circle in turn finding the value of smaller radius.
• finding area.