# No One Has Been Able to Answer This

• Roger Thomas
In summary: Finally, use Heron's formula to calculate the area of the triangle and you are done.In summary, the conversation discusses a math problem involving finding the area of the space between three circles of the same size arranged in a specific way. Multiple suggestions are given, including using equilateral triangles and sectors to calculate the area. It is also mentioned that the problem can be solved for circles with different radii using vectors and Heron's formula.
Roger Thomas
Recently, the following math problem was posed to me:

What that chicken scratch above shows is a problem I have not yet been able to solve. By what formula can one determine the area of the space in between three circles of the exact same size touching each other in the arrangement shown above?

The area in the middle isn't a straight edged triangle, it's a three sided space with each side having a curve that somehow relates to the diameter of each circle, I think.

If anyone has any thoughts on how to create a formula to solve for the middle area given any diameter for the three circles or if anyone could point me in the right direction I would appreciate it.

Last edited by a moderator:
The area would vary, of course, with the identical diameters of the circles. Trying to dig down into my rusty brain (I won't pretend any of this is from the top of my head), I would think one solution would be to draw an equilateral triangle between the centerpoints of the circles, compute the area of that triangle (call this result X), compute the area of each wedge (I believe they should be identical) formed by the two radii of the triangle intersecting the perimeter of each circle (call this Y). Someone stop me if I am mistaken here, but the points of contact between the circles (I assume they are intended to each have a radius that exactly contacts the other two circles at a single point) should lie on the triangle's laterals. If I am right on that, then I think the area specified would be X - 3Y.

Maybe you can you use integration--restricting the graph of the circle--to find the area between the graphs.

Then, choosing coordinates carefully, say the radius is r. Then center two circles at (r,0) and (-r,0), and the top circle would be centered at (0,2r), find the points of intersection.

I think that should work.

I have an idea to do this.(I assume that all the circles are of same radii)
first draw tangents at all the common points between the circles.By symmetry three tangents will cut in between the space between three circles and angle between three tangents will be 120 degrees(do you got it?).
Do you understood how i write last line??

I doesn't understand what you write in right part of image...
see attachment i have edited and drawn tangents
for different radii see the link given by chiro. that requires knowledge trigonometry.
why go for integration and other lengthy processes if it can be quite easy done with simple geometry..

#### Attachments

• Math_FindTheArea1.jpg
43.9 KB · Views: 505
Last edited:
This can be solved pretty trivially. About 2 lines.

The trick is to rearrange it into areas much easier to calculate. By tiling.

Here is a big hint.

#### Attachments

• circles.png
3.7 KB · Views: 563
Last edited:
DaveC426913 said:
This can be solved pretty trivially. About 2 lines.

The trick is to rearrange it into areas much easier to calculate. By tiling.

Here is a big hint.

it is also valid only for similar circles else it became to tricky to find out the area of quadiletral as well as of circles. am i correct DaveC426913

the triangle formed by joining the centres of circle is equilateral with side 2r. area of this is
√3/4*(2r)^2. now area of the three sectors is 3(πr^2/2).
required area is subtraction of the above two.

vkash said:
it is also valid only for similar circles else it became to tricky to find out the area of quadiletral as well as of circles. am i correct DaveC426913
The OP did specify that all circles are the same size.

altamashghazi said:
the triangle formed by joining the centres of circle is equilateral with side 2r. area of this is
√3/4*(2r)^2. now area of the three sectors is 3(πr^2/2).
required area is subtraction of the above two.
Heh. Exactly the same as my answer except I doubled it.

if it is done so simply then why double it and make it more calculative.

altamashghazi said:
the triangle formed by joining the centres of circle is equilateral with side 2r. area of this is
√3/4*(2r)^2. now area of the three sectors is 3(πr^2/2).
required area is subtraction of the above two.

When you wrote 3nr^2/2 did you mean nr^2/2?

When you wrote 3nr^2/2 did you mean nr^2/2?

Yes, it should have been $3(\pi r^2/6)$

altamashghazi said:
if it is done so simply then why double it and make it more calculative.

Yep. Yours is better.

yes it was mistakely written. i meant πr^2/2. mentallic is correct.

dijkarte said:

Care to share it?

The idea is to form a triangle whose vertices are the circles' center points. Calculate the area of the triangle and subtract the sector/pie area of each circle.

Assume the circles have radii of r1, r2, and r3.

You can position one circle at the origin p1(0, 0), the second will be at p1(r1 + r2, 0) and the trick is to find the position of the third circle p3(a, b) which is left as exercise. Now using the three positions, you can use the vectors [p1, p2], [p1, p3], ... to calculate the angles of each sector, which is then used to calculate the area of the sector.

## 1. What is the question that no one has been able to answer?

The question that no one has been able to answer is highly subjective and can vary depending on who you ask. Some may say it is the question of the meaning of life, while others may argue it is the question of the existence of a higher power.

## 2. Has anyone come close to answering this question?

There have been many attempts to answer this question throughout history, but it ultimately depends on one's personal beliefs and perspectives. Some may argue that certain philosophies or religious beliefs provide answers, while others may argue that the question remains unanswered.

## 3. Why do you think no one has been able to answer this question?

The question itself is highly abstract and open to interpretation, making it difficult for one definitive answer to be universally accepted. It also delves into complex and subjective topics such as morality, spirituality, and the nature of existence, which can be challenging to fully understand or explain.

## 4. Are there any ongoing efforts to find an answer to this question?

There are always ongoing efforts to find answers to the big questions in life, including this one. Scientists, philosophers, and religious leaders continue to search for answers through research, discussions, and debates.

## 5. Will we ever find an answer to this question?

This is a difficult question to answer, as it ultimately depends on one's personal beliefs and perspectives. Some may argue that we will never find a definitive answer, while others may argue that we are constantly evolving and learning, and may eventually find an answer in the future.

• General Math
Replies
2
Views
3K
• General Math
Replies
2
Views
1K
• General Math
Replies
2
Views
1K
• General Math
Replies
1
Views
815
• General Math
Replies
6
Views
1K
• Engineering and Comp Sci Homework Help
Replies
2
Views
931
• General Math
Replies
8
Views
4K
• General Math
Replies
7
Views
2K
• General Math
Replies
3
Views
1K
• General Math
Replies
1
Views
2K