Find the Blue Area

andymath.com/geometry-challen...
For more geometry challenges check out the above link!

[@mrmemer4589]

I’m going to take off half a point from this question on the test because the unit is centimeters was included and you didn’t specify in your answer

[@AndyMath]

Yes. I deserve that half point taken off!

[@okamiexe1501]

​@@AndyMath nooo you dont 😂

[@khaelkugler]

I just had some major secondary school flashbacks

[@marvinochieng6295]

@@okamiexe1501oh my. This takes me back to my high school days when such half a point made me not qualify for the national math contest. Thank you for making me feel that memory all over agian

[@Yu28_]

​@@khaelkuglerOh, aren't you the one who plays 3×10^8 ms^-1?

[@brbailey]

The rotating of r1 and r2 to form the hypotenuse blew my mind. I gasp aloud.

[@dartingralaughter9781]

Lol. It's funny, I did that but I made the side x but messed myself up on the calculus

[@tomdekler9280]

Presh Talwalker always makes it a point in his videos to highlight that if two circles are tangent, their radii are colinear and touch at the point of tangency. Simply said, if two circles touch eachother, there is a straight line that goes from both midpoints of the circles through the point where the circles touch.

[@ColonelBummleigh]

It was groundbreaking.I could almost hear the IMAX music when I watched it. The animations are great.

[@DlcEnergy]

Yep, that's where the solution was hidden. Bringing those radius's together, then using pythagoras to get us an equation which very easily reduced to r = √2. And then plugging that in the simple equation for the blue area which easily simplified down to π*3. Lets put that in a box. [π*3] How exciting.

[@blaAbB]

yea same. I tried it before and gave up but when he rotated r1 and r2 to form the hypotenuse i literally fell off my chair.

[@esunisen3862]

Unit of rectangle side: cm Andy: We don't use this in the US.

[@bangsulthon]

Man, i randomly found your channel and now i'm addicted to it... It's remind me the fun of math that i forget decades ago... Massive thanks...

[@hoonie8650]

🎉

[@FirstPersonLife]

It’s almost like a puzzle, I like to think of how to solve them without actually solving them. In this example I figured out that once I had that right triangle I’d be able to solve for r and get the solution. The problem is, after that, I don’t know if I’d have done the actual algebra correctly if I had a pen and paper in front of me.

[@MiniNoahTheWorm]

I didn't think connecting the two radiuses to form the triangle was an option because I couldn't convince myself that they would perfectly align

[@howmanybeansmakefive]

I thought the same thing. But then considered, where the two circles are just touching each other, there can only be one gradient for the line of tangency, for both circles at that point. i.e the tangent lines must be equal. Try to imagine the tangent gradients somehow being different and you can see why that doesn’t work. If they share the same tangent line, then the radii have to both be perpendicular to the (same) tangent line, so the radii must be 180° (i.e. a straight line)

[@bpetrushev]

This always happens when two curves (eg. circles) are touching at a single point

[@user-tw8mm9nc3j]

​@@howmanybeansmakefive thank you man.

[@JohnnyOU08]

I love this channel. There is zero fat on these videos. You just shred through these math problems. Keep 'em coming!

[@Kaget0ra]

What a great subscription your channel is! I love these fun little easy problems, but I'd never make time for them without your new videos in my feed to remind me. The ones where your solution is quite different than mine are a special treat.

[@karl8088]

You bring so many interesting questions. Nice contents!

[@NightSkyJeff]

To save a step or two, you don't need to know the value of r{1}, you need to know the value of (r{1})^2, and you figure that out when you solve the triangle.

[@michaellacaria910]

Wish i had these animations and explanations 40yrs ago! 😊

[@Peter_Riis_DK]

Well done, but I believe the unit is cm...

[@jacklambert8112]

Cm^2 !!

[@sammafo7131]

cm²

[@tiramusi]

@@sammafo7131 Andy already wrote 3pi unit^2, so the unit is cm.

[@hjewkes]

A cool problem would be to take the Ten Penny Puzzle solution (packing of 10 circles in a square) and solve for the length of the sides of the square. Using the relationship of circles to determine the size of a square feels right up your alley.

[@kidneybeans8937]

at 2:14 you already have the value for r1^2. you could have just plugged it in to 3pi(r1)^2/2. No need to take the extra step of square rooting it, then plugging it back in only to square it again. Great vid BTW

[@user-nw1lo6cq7i]

which software do you use to animate and present the equations???

[@jobaecker9752]

Got this one...but with a slightly different approach. How exciting.

[@alineharam]

I look at these problems, but then I do not know how to solve them, yet the solution, if you follow rules of reason, appears. Interesting ideas about 'problems' in general. When will we follow the 'rules' for reasoning to solve our problems? And the answers are not obvious. I love the Andy Math channel.

[@muhammadfaris5542]

How do you know the hyphotenuse of the triangle will intersect (touch) the center of the two circles? In other word, how do you know R1 and R2 can be in straight line to form a hyphotenuse for the triangle? Thank you.

[@tomdekler9280]

Circles that are tangent to one another have colinear radii on the point of tangency. There's several proofs for this. The shortest distance from a point to a line is a perpendicular line. This is why lines tangent to a circle are always perpendicular to the radius; if there was a shorter distance, the line would intersect the circle. At the point of tangency of two circles, the perpendicular lines match, so they're colinear.

[@djdynieldaniel1395]

This look important. Let's put a square around it.

[@Paradox-yt2718]

Can you please tell me what software you are using?? Want to follow in your footsteps

[@CrazyChemistPL]

Actually you can label it as 3pi cm^2, since they flat out give us the length in centimeters.

[@donaldtier7395]

May I ask why r1+2r1 must be a strangle line? I.e. the additional line linked centre of the r1 semi-circle and the right bottom corner of the rectangle may not cut the intercept point of the two circles. Thanks.

[@starboy001]

What and which University have you studied in Andy?

[@robertthe19th]

That was a really clever way to solve it.

[@samuelking4723]

How do we know for sure that the r1 + r2 hypotenuse is actually linear/the angle between the connected radii is 180 degrees? Is there some mathematical principle that if two circles are tangent to each other, the line between their centers will necessarily run through the point of tangency?

[@craigheimericks4594]

Yes. All tanget lines of a circle are perpendicular to the radius at that tanget. If two circles share the same tanget line, their radii are both perpendicular to that line and thus they must be 180 degrees.

[@samuelking4723]

@@craigheimericks4594 Well shit, I can’t believe I didn’t think of that. Thanks man.

[@ramnaftaliavni6532]

When did he mention 180 degrees? I've missed it, and I want to understand the question, and the answer that nice guy gave you :)

[@AWhistlingWolf]

Draw two circles of any size each, put a straight line from the center of one to the center of the other and make the circles' borders touch. You won't find any way to do it without making the line and the point of touching meet.

[@samuelking4723]

@@AWhistlingWolf Yeah I got that, the question was how do we actually know that for certain? Another commenter already explained it.

[@GrahamMiao]

To those who wonder about the rotation part, it can be proven that the centres of the tangent circles and the point of tangency are collinear.

[@Filsduberry]

What a wonderful channel . Thank you so much

[@naalex9058]

I did it in my head! Watching your videos made me become a *god*

[@Felipera_]

I've been wathcing these too much, rotating the radiuses to form a hypotenuse was my first thought lol

[@memestrous]

I havent watched the full video yet but I got 3pi. I first drew a line through the centres of both circles. That would form a right triangle with radius of smaller circle (x) and the base 4. Then you can form an equation like 2xy + y² = 16. And we can clearly see that the right side of the radius of the larger circle (y) and the left side is the diameter of the smaller circles (2x). So y = 2x. Then we solve them simultaneously and find the values of x and y and use them to find the area which is 3 pi

[@jaye1967]

Given that we start with the rectangle having a side of 4 cm, I would say the correct answer is in cm as the units are known. I'm not sure if this would just be nit picking on my part, but it just feels more correct.

[@johncarrington8612]

Bonus points to do the problem with calling the quarter circle radius r and the semicircle radius r/2. The solution is the same but the work is harder.

[@DChiru2527]

Thanks man im gonna challenge my teacher 😂 ❤❤

[@yepyepmusic]

How exciting? Really exciting!

[@garnekux123]

Ok but how do you know at first place that these are semi-circle on the left and quarter circle on the right?

[@bushwalker6214]

Solved it by just staring at the thumbnail of the video w/o opening it. A nice task for a last 2-3 years of high school.

[@DylanCyr]

How do you know the lower right corner is the center of the large circle?

[@gouriss]

How do i send you questions?

[@sbmlemos]

How can you ensure that the two rays will be aligned and form the hypotenuse?

[@majdsait3236]

I was working on it and got to the point where I'm not sure whether it's possible or not to draw the (r1 + r2) straight line...can you please explain how is it possible...😁

[@MrFrmartin]

not too bad of a question. was able to solve in double the time

[@Its_just_me_again]

im not flexing here, but i never did math at school and even before you explained the answer i KNEW i had no idea

[@memesalldayjack3267]

I'm not smart enough to confirm whether those answers are correct, I'm also not in the mood to pay very close attention to it all, but i like watching it, cause i see small things that'll probably help me someday, like thinking about making a triangle with that radius

[@marvinochieng6295]

i like how well articulated you are.

[@memesalldayjack3267]

@@marvinochieng6295 thanks, I've spent a lot of time practicing

[@D-Dova]

Neither am I but I genuinely beleive thw answers arnt correct hes pulling out nonsense from nowhere I like watching his videos and id like to learn but it just seem true the answers

[@DandoPorsaco-ho1zs]

I got the same answer.

[@kellyeaton7252]

How exciting!

[@cameronhauberg5026]

How exciting indeed!

[@jawanen]

I dont like this kind of math questions because tou have to assume those are circles. Better if it was stated

[@TheTallRaver]

1:35 - how do we know that these two radii create a straight line to form a triangle? It looks like you just assumed it but is there a rule that explains/confirms that? How do we prove that?

[@TheTallRaver]

Ok, someone else expalained that below😉

[@ilyakovalyov5751]

Why do you assume that r1+r2 makes a triangle? What is the proof for that? It seems very convenient but how can you actually prove that the connection between large and small circles lays between the centers of both circles?

[@ccdsah]

u did an extra step, u didn't need to find r1, u just need r1^2

[@xa-69prototype-19]

Don't take sq root of 2 then square it substitute directly

[@Yuuichi399]

Looks Hard Simple after seeing him solve Yet How Exciting❤💯

[@MatheusLB2009]

How can you be sure the pythagoras aligns nicely?

[@user-dc3bv7bk7x]

Why these 2 sectors don't overlap when the hypotenuse equals to r1 + 2*r1? Why r1 and 2*r1 must bond to a straight line?

[@evetheeevee2977]

Insert filler text claiming spoilers Let the radius of the semicircle be x. The radius of the quartercircle is twice the semicircle, so its radius is 2x. Drawing a line connecting the centers of the circles, we can use the Pythagoras Theorem to derive the following equation: 4² + x² = (x + 2x)² 16 + x² = 9x² 8x² = 16 x² = 2 pi(r)² ½pi(x²) + ¼pi(2x)² = ½pi(x²) + ¼pi*4x² = ½pi(x²) + pix² = 1½pix² = 1½pi(2) = 3pi ≈9.4247

[@7hephi779]

Im curious how one can find that r1 and r2 can make that triangle? is there a proof of it somewhere?

[@miguelangelrosascastillo3776]

You can draw a line between the radius points of two circles that are touching and it’s going to be equal to r1+r2

[@joaocurado5277]

Theres a little where both circles are the closest to each other indicating that theyre touhcing, therefore you can draw a line through it connecting to the corner of the square

[@eeple29]

A radius will always be perpendicular to a tangent line. Since the semi circle and quarter circle are assumed to be touching at only a single point they will have the same tangent line at that point. Since each radius is perpendicular to the same line they must have the same slope and therefore make one continuous line

[@xNillowsx]

think of the motion of the tip of a minute hand of a clock over an hour, then simplify that back to 2D (you should be imagining a perfect circle). you know for a fact this circle is defined by the length of the minute hand (radius) you imagined and at all points on the curved line the minute hand remained constant in length. now put another "clock circle" directly beside it, but with a smaller radius and circumference. in this analogy you can imagine that if the 2 circles should overlap, then at some point in time the hands will collide with each other, and would need to be spread further apart in order to keep proper time. this example has the circles TOUCHING so imagine perfectly positioning the clock so the hands form a perfectly straight line. the proof comes from the acknowledgement that the radius is a fixed property of any natural circle.

[@7hephi779]

@@eeple29 This is a great way to explain it. Thank you!

[@shpingaletty]

I'm assuming there should be no assumptions in STEM fields

[@SaiHikawa]

So... This might be in the assumption that the edges of the rectangle is the diameter of the smaller circle and the radius of the bigger circle... But what if it isn't? Like, if the half circles are not entirely half and quarter?

[@kylewatson5133]

How can you assume that is a quarter or semi circle? If you added a single pixel in illustrator there is no way to tell. It is just given in the description of the problem?

[@paper-claws9961]

How exciting.

[@krobro1627]

I can see it, its right there

[@Joshua_23]

How exciting 💯💯💯🔥

[@jackdean2509]

how exciting indeed

[@shaylevinzon540]

That's the blue area 🎉

[@MrMichaelBradfield]

did this whole thing in my head in a few minutes... 3Pi (x/2)^2 + 16 = [ (x/2) + x ]^2 --> x = 2sqrt(2) blue area --> area of a circle is Pi x radius ^2 --- radius of semicircle is 2x the radius of the quarter circle do the math and you get 3Pi

[@dr.adityaguptadentist1799]

I am not capable to understand such level of math

[@annoyingbstard9407]

I found it. There’s a bit on the left and a bigger bit on the right.

[@boabnr418]

Well that was easy, it is a good problem definitely

[@ravikiran8278]

How did you assume that the the center of smaller circle, the center of larger circle and their point of contact lie on the same line?

[@7212372frank]

I agree.

[@kimjongunvevo]

At any point on a circle, there can be only one tangent at that point and it will be perpendicular to the radius. Both circles meet at one point, so both circles have their respective tangents on the same point, and both the tangents will be in the same line, then obviously the two radii will also be in a straight line.

[@eroticbearvideos]

try to make two circles NOT line up that way

[@BaldZippy]

Making the triangle is very creative I didn’t think of that good video

[@NatashaRomanoffThailand]

Why is the unit of the answer is u^2? Should it be cm^2 because the picture already shows 4cm?

[@gaming_odyssey5399]

It should be, he made a mistake

[@aszx-tv4pq]

Although the correct answer is 2, you should have demonstrated that the segments are aligned with each other, as we are uncertain about where the arches intersect. But very fun and interesting content 🎉🎉🎉

[@eroticbearvideos]

circles are always going to meet between their centers

[@robertoreyes2503]

Proud to have solved this in my mind

[@sharp_tooter]

1:16 wait, how do the 4s just magically turn into 2s? is it just because they cancel each other out so you could make them any old number and it doesn't matter?

[@sammafo7131]

True right answer is : ? = 3π cm²

[@kaiji2542]

I think it's because he often works on size problems involving numbers only without any unit of measurement. I have seen a comment pointing out this mistake and he hearted the comment as acknowledging his mistake

[@jjemluuu6336]

it's not unit square tho, cause one of the rectangles length is literally 4cm

[@rasmuspedersen3563]

where does those divided by 2 and 4 come from? I just curious im not a math geni...

[@tastyl2356]

the area of a circle with radius r = pi*r^2. the shape on the left is half a circle, so to get its area, we use the formula for area of a circle, but then half it. same logic applies to the right shape, which is a quarter of a circle, meaning we divide it by four.

[@lolok6439]

the idea for rotating the radii, thats genius

[@tomdekler9280]

It's the only way to solve. The width of the rectangle is entirely dependent on the two circles touching. Without it, the 4cm that we were given wouldn't do anything.

[@lolok6439]

@@tomdekler9280 i know it is, but i wouldve just went around in circles (lol) and never wouldve gotten anywhere

[@tomdekler9280]

@@lolok6439 yeah the fact that circles in tangent have colinear radii is a huge part of most circle-related geometry problems.

[@JDP1977]

How exciting 💯💯💯🔥🔥🔥💀💀💀

[@secretnobody6460]

Find the blue area Me: there

[@redpandagalaxy2806]

How exciting

[@zanmatoshin877]

I found the blue area, it's towards left and bottom right. Inside a rectangle

[@FlauxT]

I tried so hard to find the triangle in this and couldnt' do it :(

[@akinamegu9896]

people like you are great and awesome ! youtubers should follow your example for you have done right by seeking to provide good and educational content for students ! you and another good man like yourself known as "presh talwalker" are what a youtuber should aspire to be !

[@AlmightyFSM]

ok that was good

[@alexandergooding4860]

What if the question stated it was not a quarter circle would it be possible mathematically (assuming the diagram is not to scale)?

[@tomdekler9280]

Only if it's some other defined polynomial. For instance, this question would be easier if the right shape was a square instead of a quarter circle. The square would be (2r)² and the half circle would be πr²/2 The only stinky bit is that r² would be 16/9, bit of an ugly number.

[@Genebriss]

The question did not state that it was a quarter and a semi circle so the task in the picture is actually impossible to solve. You have to complete the task by assuming that they are quarter and semi circles.

[@donkeykong1974]

01:39 It's not obvious. U'd prove it.

[@romypotash7114]

1:39 i had to puse to relize why it's true (both are vertical to the tangent of the circuls in that points, with needs to be the same)

[@petrouvelteau7564]

Is it just me or does him saying "r sub-one" sound like "arse of one"?

[@ThanatoselNyx]

Thank you, I was hearing Arse Of One and couldn't figure out what he was actually saying!

[@Noor-sl5ep]

Funny symbols you have in there, magic man.

[@aarusharya5658]

You must be really young then

[@Noor-sl5ep]

@@aarusharya5658 last year in highschool

[@Noor-sl5ep]

@@aarusharya5658 +ngl it does look very confusing

[@archanatripathi2534]

Love from India 🇮🇳 sir

[@mikebrandon498]

units are cm

[@saucahr]

I am in 9th grade and average at math. Idk how but I solved this first try. Am I good at math now???

[@W.2026]

How do we know the three dots in 1:37 make a straight line though?

[@Oleg50600]

I'm wondering the same thing

[@KaxMerg]

it is implied BECAUSE there is a dot, similar to how lines are often implied to be parallel by dashes through them

[@liluxiaorealnew]

Straight line is shortest distance between two point.

[@prestonnewcomb5991]

Ok, how can one assume that is a half circle and quarter circle.

[@hiftu]

Maybe this solution would fly in math, but not in engineering. Unit squared? Only unit I've seen is cm. Your u^2 can be any imaginary thing, unrelated to this problem.

[@Ayvengo21]

How do you know that on 2:07 r1 + 2r1 line is straight line and not the curved one? Guess it's required to solve this task but how to prove this assumption.

[@isaacmiddlemiss7790]

Think of it this way: if it was not a straight line, ie it was bent, that would mean that you could straighten it and have it be longer. However, the start points of each line are fixed, so the point where the two arcs touch must be when the line is the longest it can be, ie when it's straight

[@TheMidwestGaming]

I'm here within 3 minutes of uploading! How. Exciting.

[@user-zp9oi3cw1m]

This is easy grade 6 math!

[@ritmikus_csimpifon3544]

unit square should have been cm square

[@jeremiedsouza6703]

How do you know that r1 and r2 form a straight line at 1:36 ?

[@Atulya_YT_]

By using your brain

[@kaiji2542]

All tangent lines of a circle are perpendicular to the radius at that tangent. If two circles share the same tangent line, their radii are both perpendicular to that line and thus they must be 180 degrees.

[@jeremiedsouza6703]

@@kaiji2542 ohh I see. Thanks!

[@mikewillis1592]

This is not solvable without making an assumption that there are half and quarter circles.. They look like they may be, but it is an assumption.

[@ColinRichardson]

I'm not bothered enough to know the number I just like to know the theory behind finding out.. The whole "reducing down to the exact number" I would just let the computer do.. I just need to know what numbers to plug in.