An Awesome Wave

Geometry Level 2

The blue figure is made of five quarter-circle arcs (each with integer radius) and a segment of length 4, which is the base of a 4 × 4 4\times 4 square grid.

What is the area of the blue figure?


The answer is 5.71238898.

Andrew Hayes by Andrew Hayes

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

6 solutions

Andrew Hayes Staff
Apr 17, 2018

You can construct the shape by adding/subtracting quarter-circles one-at-a-time.

Adding/subtracting these areas gives 3 2 π + 1 ≈ 5.712 . \frac{3}{2}\pi+1 \approx \boxed{5.712}.

142 Helpful 0 Interesting 0 Brilliant 0 Confused

Is it Square 1 - Square 2 + Square 3 - Square 4 ?

Chakravarthy R - 3 years, 1 month ago

Log in to reply

Yes -- you can see which parts are added or subtracted outlined in red. I've used solid red outline to indicate that it was added and dashed red outline to indicate that it is subtracted.

Andrew Hayes Staff - 3 years, 1 month ago

Why are you adding in that 1 x 1 square ?

John Conway - 3 years, 1 month ago

Log in to reply

It is part of a quarter-circle that needs to be subtracted in the next step.

Andrew Hayes Staff - 3 years, 1 month ago

Log in to reply

@Andrew Hayes – Yes, thank you Andrew - it dawned on me shortly after I posted that comment, lol .

John Conway - 3 years, 1 month ago

the web does not accept 5.712, it somhow showld tell the amount of decimals it require to accept the answer

Genaro Canero - 3 years, 1 month ago

Log in to reply

Our records show that you entered 5.172.

Andrew Hayes Staff - 3 years, 1 month ago

May I ask what program did you use to generate the images? :)

Alan Yan - 3 years, 1 month ago

Log in to reply

I want to know that too. I approached this problem differently and want yo add this as an answer.

Sarvil Pargi - 3 years, 1 month ago

I had the solution with π and the program didn't accept it. What a disappointment!

Olivier M - 3 years, 1 month ago

Respected Sir and everyone , Please excuse me for posting this comment . Rather than saying that this is a comment this a burning issue in my heart . Please take it seriously .

Note : This post is regarding the Easy section of this problems of week .

I have a dream getting 100 or more up votes for an individual solution . I wrote a solution for the question water shadows (which is adjacent to this question) 6 days ago (on 21 April) . When I wrote the solution I thought it is a good one and will definitely accomplish my dream .

I am extremely happy when I saw that the question is in one of the problems of the week . I thought I am nearer to my dream because nobody posted any solution at that time expect me to this question and my solution is a clear and good one . But after these four days I am extremely sad because even more than 11,000 people solved that question only 83 people are discussing solutions . As a result at present I am only left now with 45 up votes . Although my solution deserves more than 100 up votes due silly reasons it is unable to accomplish my dream .

At first when I solved this question (infinite squares) on Monday morning I solved in the manner you did . I thought to post the solution but when I saw you have already posted the same solution (and now you got 299 up votes) . But I didn't worry at that time because I had hope that my solution to water shadows question will definitely cross 100 up votes .

Once see the up votes to the top solutions for the 5 questions of the problems of this week and see how odd it looks :

Infinite squares - Jason Dyer - 349 up votes

Water shadows - Ram Mohith - 57 up votes (see how odd it looks)

Third problem - Zain Majumder - 167 up votes

Fourth problem - Jeremy Galvagni - 129 up votes

Fifth problem - Micheal Mendrin - 250 upvotes

My point is that please try to discuss solutions to every question you solved and try to up vote the solution you admire as much as possible . If you do that many people like me will acheive what they deserve .

Please try to understand my problem .

Ram Mohith - 3 years, 1 month ago

37 Helpful 0 Interesting 0 Brilliant 0 Confused

This solution makes complete sense to me. Thank you Marvin Kalngan .

John Conway - 3 years, 1 month ago

This solution makes sense. What if we some how develop equations of these curves integrate and add them. How to develop ones??

Awais Zafar - 3 years, 1 month ago

Respected Sir and everyone , Please excuse me for posting this comment . Rather than saying that this is a comment this a burning issue in my heart . Please take it seriously .

Note : This post is regarding the Easy section of this problems of week .

I have a dream getting 100 or more up votes for an individual solution . I wrote a solution for the question water shadows (which is adjacent to this question) 6 days ago (on 21 April) . When I wrote the solution I thought it is a good one and will definitely accomplish my dream .

I am extremely happy when I saw that the question is in one of the problems of the week . I thought I am nearer to my dream because nobody posted any solution at that time expect me to this question and my solution is a clear and good one . But after these four days I am extremely sad because even more than 11,000 people solved that question only 83 people are discussing solutions . As a result at present I am only left now with 45 up votes . Although my solution deserves more than 100 up votes due silly reasons it is unable to accomplish my dream .

At first when I solved this question (infinite squares) on Monday morning I solved in the manner you did . I thought to post the solution but when I saw you have already posted the same solution (and now you got 299 up votes) . But I didn't worry at that time because I had hope that my solution to water shadows question will definitely cross 100 up votes .

Once see the up votes to the top solutions for the 5 questions of the problems of this week and see how odd it looks :

Infinite squares - Jason Dyer - 349 up votes

Water shadows - Ram Mohith - 57 up votes (see how odd it looks)

Third problem - Zain Majumder - 167 up votes

Fourth problem - Jeremy Galvagni - 129 up votes

Fifth problem - Micheal Mendrin - 250 upvotes

My point is that please try to discuss solutions to every question you solved and try to up vote the solution you admire as much as possible . If you do that many people like me will acheive what they deserve .

Please try to understand my problem .

Ram Mohith - 3 years, 1 month ago
Toxast .
Apr 23, 2018

34 Helpful 0 Interesting 0 Brilliant 0 Confused

I don't really understand this ?????

John Conway - 3 years, 1 month ago

Log in to reply

One green area is missing in the original, whilst one is in the original, so he's moved the left hand green area into the gap where the right hand green area is. Similarly with the yellow areas.

Dan Ley - 3 years, 1 month ago

The question calls for a numerical solution. You've certainly set that up, but....

Richard Desper - 3 years, 1 month ago

Nice method to solve

Kshitij Pahwa - 3 years, 1 month ago

Fun question, but you could have made it a little more clear what you meant by "made of quarter-circle arcs." An arc is just a curved portion of the circle's circumference, i.e. a curved line. You weren't adding or subtracting curved lines from the blue figure; you were adding and subtracting quarter-circles.

Daniel Jackson - 3 years, 1 month ago

add a quarter circle with r=4, subtract a quarter circle with r=3, add the white space of a quarter circle with r=1

(4^2/4)π - (3^2/4)π + (1 - (1^2/4)π) = 1 + (6/4)π = 5.7124

Mira Jain - 3 years, 1 month ago

I thought they meant each segment was 4 making the radius of the big circle 16. I would have had it correct if not for that...a clearer picture showing the segment length on the drawing would have helped.

Brendan Lennon - 3 years, 1 month ago

Log in to reply

The question clearly stated it was a 4 by 4 though...

Timothy Low - 3 years, 1 month ago
Ravi Kant
Apr 27, 2018

5 Helpful 0 Interesting 0 Brilliant 0 Confused

I CONSTRUCTED THE SAME THING IN SOLID WORKS

FREDERICK ROSE - 2 years, 9 months ago
Regis Kopper
Apr 26, 2018

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Robert DeLisle
Apr 26, 2018

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...