Find the area of the marked in black

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ksardas21

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1617423620558.png
In a circle with a radius of 2+√3, three circles with the same radius are drawn
so that they all touch each other and the big circle (as in the picture).
We need to find the area (S) marked in black between the three circles.
 
ksardas21 said:
View attachment 26143
In a circle with a radius of 2+√3, three circles with the same radius are drawn
so that they all touch each other and the big circle (as in the picture).
We need to find the area (S) marked in black between the three circles.
Click to expand...
What thoughts do you have? Have you tried anything? We need to know what help you need.

I would start by drawing in radii to all the points of tangency.
 
ksardas21 said:
That's what I triedView attachment 26146
But I have no idea what to do next
Click to expand...
Then try something totally random! The important thing is to try something, rather than sit there as if you are totally helpless. You don't need to wait for someone else to do things for you.

There are more radii you can draw -- radii of the larger circle!

Once you've done that, think! Tell us some things you observe, even if you think they are useless.
 
Dr.Peterson said:
Then try something totally random! The important thing is to try something, rather than sit there as if you are totally helpless. You don't need to wait for someone else to do things for you.

There are more radii you can draw -- radii of the larger circle!

Once you've done that, think! Tell us some things you observe, even if you think they are useless.
Click to expand...
1617429128675.png
I am here right now
 
ksardas21 said:
View attachment 26148
I am here right now
Click to expand...

Okay, now tell me what thoughts you have.

Think about what your goal is: How might you find that black area? Maybe by subtracting something? What will you need to know in order to to that? (In other words, work backward, looking for possible last steps.)

Think about what more you can determine from what you know: Maybe you can find a radius or two? Maybe the sides of the triangle? (In other words, work forward, looking for possible next steps.)

Think about how you can get from what you know to what you need to know.

This is how you solve a problem: one step at a time. Take a step, and we'll try to catch you if you fall.

By the way, you still haven't drawn any radii for the large circle! That may be necessary, though in fact you have used an implication of that radius already.
 
Your geometric drawing is beautiful. I admit I liked it. If you can figure out the radius of one of the small circles, you are done. It just needs a little of thinking outside the box.
 
Dr.Peterson said:
Okay, now tell me what thoughts you have.

Think about what your goal is: How might you find that black area? Maybe by subtracting something? What will you need to know in order to to that? (In other words, work backward, looking for possible last steps.)

Think about what more you can determine from what you know: Maybe you can find a radius or two? Maybe the sides of the triangle? (In other words, work forward, looking for possible next steps.)

Think about how you can get from what you know to what you need to know.

This is how you solve a problem: one step at a time. Take a step, and we'll try to catch you if you fall.

By the way, you still haven't drawn any radii for the large circle! That may be necessary, though in fact you have used an implication of that radius already.
Click to expand...
It is enough to know the radius of small circle, and S(big circle)-3*S(small circle)= S(black area)
yeah my main aim is to somehow find the radius of small circle and I am thinking about the ways of solving this problem
 
ksardas21 said:
It is enough to know the radius of small circle, and S(big circle)-3*S(small circle)= S(black area)
yeah my main aim is to somehow find the radius of small circle and I am thinking about the ways of solving this problem
Click to expand...
Not quite; if you subtract the small circles from the large circle, you will be left with more than just the black area.

But you might subtract three sectors of circles from the triangle ...

In any case, yes, finding the radius will be a major step. If you show us some of the ideas you have, we can probably adjust one of them to make it work.

I would focus on the equilateral triangle; what do you know about its altitudes/medians? This will be where it's helpful to extend the lines you've drawn into radii, or even diameters, of the large circle.
 
Alternative for finding r
Untitled.jpg

[MATH]\text{Cos}(30) = \frac{r}{R-r}[/MATH][MATH]\frac{\sqrt{3}}{2}=\frac{r}{R-r}[/MATH]
[MATH]\sqrt{3}R-\sqrt{3}r=2r[/MATH]
[MATH]r=\frac{\sqrt{3}\text{ } R}{2+\sqrt{3}}=\sqrt{3}[/MATH]
 
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