[Guest]

The idea is this:

1. Start with an equilateral triangle with side length r.

2. The points of the triangle are the center points of three circles with radius r.

3. Around these three circles is a bigger circle that is tangent to the three smaller circles.

4. Within the three smaller circles is another circle that intersects at the points of the original equilateral triangle.

What is the total area of the shaded regions in terms of r. That is find an f( r ) which relates r to the shaded regions.

Please explain your method.

For those industrious enough, visualize this concept in 3D with an equilateral pyramid with side r as the start surrounded by four spheres. A larger sphere is tangent to those four spheres. And within the four spheres is a smaller sphere who's surface is touched by the four points of the original equilateral triangle. Now imagine the area asked for in the first part, apply this to the spheres and find an f( r ) which relates r to this three dimensional area.

Good luck every, first part is hard, second part I don't even know where to start!

This is a very rough diagram to better illustrate the question. Circles are not perfect and the triangle is not equilateral, but you get the idea:

Edited June 17, 2009 by lovelife

[Guest]

{ (pi)r² (2*(3)^1/2 - 1)/3 } - (3)^1/2*r² /2

Well, some one solve the second part ... ill see the soln

[Guest]

Let area of triangle (root3 / 4 x r²) = T

Let area of each fo three circles (pi r²) = C

Height of traingle = r x root3 / 2

Then radius of outermost circle is r (1 + root3/2) = 1.866r

Area of outermost circle then is 3.48 C

Now refer below figure:

The outer shaded region is area of outer circle - [ 3x(area of inner circle) - 3 (2 circle overlap area) + (3 circle overlap area)]

3 circle overlap area is the yellow shaded portion = C/3 - 2T

2 circle overlap area is yellow shaded portion + grey shaded portion = 2 (C/3 - T)

So area of outer shaded region is: 2.15C + 4T = aprrox 8.5 r²

For inner shaded region, area of triangle circumscribed to traingle - area of 3 circle overlap

Area of circumscribed circle of traingle = pi/3 r² = C/3

So inner shaded region area is 2T

Edited June 17, 2009 by DeeGee

[Guest]

Amarnath_Caterpillar

Final Answer (Not a derived function): Shaded Area= 1.714434 r^2 units for a 2-Dimensional Problem.

circles_around_triangle_17_june_2009.bmp

[Guest]

For the 3 dimensional part, everything I explained for 2D remains the same except that this time instead of areas, we are talking volumes... I'm not really industrious enough to calculate it so I'll also leave it for the more industrious ones.

[Guest]

Total Shaded Area is approximately 1.712r^2. Attachments show how I obtained this result.

Calculations_of_shaded_area.txt

[Guest]

Technically I was wrong because I didn't state the answer as a function.

F® =1.712r^2

Edited June 19, 2009 by rabyjrm

[Guest]

area of total shaded region is = [(2/3)*pi - sqroot(3)/2]*r^2

[Guest]

The idea is this:

1. Start with an equilateral triangle with side length r.

2. The points of the triangle are the center points of three circles with radius r.

3. Around these three circles is a bigger circle that is tangent to the three smaller circles.

4. Within the three smaller circles is another circle that intersects at the points of the original equilateral triangle.

What is the total area of the shaded regions in terms of r. That is find an f( r ) which relates r to the shaded regions.

Please explain your method.

For those industrious enough, visualize this concept in 3D with an equilateral pyramid with side r as the start surrounded by four spheres. A larger sphere is tangent to those four spheres. And within the four spheres is a smaller sphere who's surface is touched by the four points of the original equilateral triangle. Now imagine the area asked for in the first part, apply this to the spheres and find an f( r ) which relates r to this three dimensional area.

Good luck every, first part is hard, second part I don't even know where to start!

This is a very rough diagram to better illustrate the question. Circles are not perfect and the triangle is not equilateral, but you get the idea:

"lovelife" can u describe the second question clearly (3D)

[Guest]

"lovelife" can u describe the second question clearly (3D)

In the 2nd part of the question "r" is the side length of an equilateral pyramid. As such, instead of three circles with radius "r", there would be 4 spheres of radius "r" at each point of the pyramid. Then the outer circle in the original would be a sphere tangent to the 4 inner spheres and the inner circle in the original would be a sphere tangent to the central pyramid.

I don't have a program, nor do I know how, to draw this in a way it would make sense, you just have to imagine it. The area in question would be the same, only in 3D there would be 4 inner areas and 4 outer areas. I think this requires quite a bit of calc, I'm not sure how else you could solve for it. I am definitely not far enough in calc to find a solution. Good luck to anyone that tries!

Edited September 16, 2009 by lovelife