Find the Area of the Green Circle if Both Small Circles have an Area of 1 | Step-by-Step Explanation

Calculate the area of the big green circle if the area of the two smaller circles is one and external tangents are drawn from the center of one of the small circles. Quick and simple tutorial by PreMath.com

[@mushfikaikfat]

Such a *cool* and a little bit *tricky* 😅 *geometrical Problem* with a elegant solution!!! *Thanks* for posting the video 😊😊 It's really helpful *sir* 👍keep going 🌿🌿🌿

[@mathsandsciencechannel]

That is the beautiful thing about maths. It's widens your mind to lots of possibilities.

[@PreMath]

Thanks Ikfat for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃 Love and prayers from the USA!

[@qbertq1]

As soon as you see the small triangle has short side = r and hypotenuse = 2r, you know you have a 30-60-90 triangle and therefore the large triangle is also a 30-60-90 triangle allowing you to equate R to 3r a bit more quickly.

[@MrPaulc222]

Yes, but I tried it before morning coffee :)

[@ΚΑΝΑΒΟΣΜΑΝΩΛΗΣ]

Dear sir, I'm sure that by constracting the shape, which is a circle with center on tangent point between B and C and R equal to that center and point A we get the solution. By that we get the center of the large circle and the two tangent points D and the "mirror" one. So R=3r, and E(R)= 3^2*r=9*1=9 without long calculations. I hope its clear. Thanks! Emmanuil Kanavos

[@barttemolder3405]

Shortcut. The circle around B has the same radius as the circle around A that touches that circle. The same is true for the circle around C. We know where that touches a similar sized circle around A, namely on the rightmost point of the circle around B. The distance from A to that point is 3r, so the circle around C also has radius 3r, so its surface must be 9 times as big, ergo 9.

[@WhitentonMike]

We know 2 circles are always similar. Once we know the ratios of the radii, the ratios of the areas are the square of the ratios of the radii. Once we determine R/r=3 then we know A/a=(R/r)^2=3^2=9. A/1=9 so A=9.

[@philipkudrna5643]

A nice combination of various fields! Very nice! Thank you!

[@PreMath]

Thanks Philip dear for the feedback. You are awesome 👍 Take care dear and stay blessed😃

[@plamenpenchev262]

Your solution is more elegant than mine. First I found AE then ED. Jumping to the end to check my answer (9) I saw the drawing ... 2r/(3r+R) = r/R. That is because I used the initial picture (stopped video) and piece of paper.

[@MrPaulc222]

I calculated the ingredients but didn't manage to translate them to calculate R. I figured out r=(1/sqrt(pi)), the triangles' similarity, and even the 30/60/90 property. I imagine those parts might have secured *some* exam marks, but not the full amount due to the absence of a final calculation. Aaaargggghhhh! That one was frustrating :) . Thanks for showing a cleaner way.

[@JLvatron]

Very nice! I originally set out to calculate the small circles radius, but your solution is much cleaner.

[@paulm.8660]

Same here. I knew we were going to have to use the area given to calculate a radius eventually, and after all that setup with the similar triangles it was so satisfying to see him substitute the whole (pi*r²) term for 1, really clever.

[@edyap2034]

I LOVE MATH....I LOVE PREMATH 👍👍👍

[@PreMath]

Thanks Edy for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃 Love and prayers from the USA!

[@malhislearninghub1069]

Sir your videos are really fabulous i watch many of them

[@PreMath]

Thanks dear for such an elegant feedback. Thank you so much for taking the time to leave this comment. You are awesome 👍 Take care dear and stay blessed😃

[@chrisboyne5791]

AFB is right angle, AB is 2r, FB is r. So AFB is 1/2 of an equilateral triangle. ACR is similar. So 2R = R +3r. R = 3r. So area of big circle is 9.

[@EnnioPiovesan]

In triangle ABE AB=2r and BE=r so ABE is half an equilateral triangle (and BAE is 30°), but also ACD is half an equilateral triangle so AC=2 CD so 3r+R=2R --> R=3r. [...]

[@ddandrews6472]

" the area of the two smaller circles is one " means the area of both circles adds to the value of 1. You solved the above by assuming each circle area is equaled to one.

[@deepasingh5457]

Thank you for clear explanation 🍻

[@242math]

tried Pythagoras but it was too cumbersome, similar triangles is the way to go as demonstrated, great job

[@PreMath]

Thanks my friend for the feedback. You are awesome 👍 Take care dear and stay blessed😃

[@MsMostanas]

Thank you Sir.you are amazing.

[@procash1968]

Amazing !!

[@beeruawana6662]

Very good question sir.thank you sir

[@ViMi1]

Where π disappeared?

[@mathsandsciencechannel]

Nice video sir. Love it. Hey there. I love challenging solving calculus,geometry,algebra and trigonometry question here on my.....

[@mushfikaikfat]

R u from India?

[@mathsandsciencechannel]

No please

[@mushfikaikfat]

@@mathsandsciencechannelWhat is your nationality?? Can you give me a chance to know this ?

[@mathsandsciencechannel]

Ghana

[@mushfikaikfat]

@@mathsandsciencechannel Good 👍 Glad to hear 😊😊 Love for Ghana 🇬🇭 ❤

[@malhislearninghub1069]

Fabulous work

[@PreMath]

Thanks dear for the feedback. You are awesome 👍 Take care dear and stay blessed😃

[@r.markclayton4821]

A rather complex proof. I looked at the diagram and reasoned that if the circle (radius r) around B is also one radius from the point A then the second circle is 3r from point A and ergo its radius is therefore must be 3r and its area 9. I did get side tracked a little as the first thing I noticed was that the angle FAE is 60 degrees, but that is irrelevant to the answer.🙃

[@nirenjanthebest2709]

I saw your video 13th Still it is awesome

[@PreMath]

Thanks Nirenjan dear for the feedback. You are awesome 👍 Take care dear and stay blessed😃

[@shreyanshpatel9740]

I got it right!!😁😁

[@charlesbromberick4247]

I like this problem.

[@India-jq7pi]

Thank you sir

[@nirmalajagdish8901]

Vvnice true thanks

[@nigelsw55]

I used exactly the same method.

[@theoyanto]

I spotted the easy way ... Go me !!🤓

[@jonathancapps1103]

1. I suspected R = 3r from observing the diagram, but it was nice to see the proof. 2. The given was somewhat ambiguous. I wasn't sure if *each* small circle had an area of 1, or if their total combined area was 1.

[@JSSTyger]

Area = 9

[@SuperYoonHo]

i like geometrical Problem

[@SuperYoonHo]

s

[@akamai897]

I got one! LOL

[@ณรงค์ฤทธิ์พลชาย]

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