Circles and triangles

  • Thread starter darkmagic
  • Start date
  • #1
darkmagic
164
0

Homework Statement



Please see the attached figure
The radius of the biggest circle is 10.
The required is the sum of all circles and the sum of all triangles in the figure.
There is an infinite number of circles and triangles.

My answers are:
for circle: 475/3 pi
for triangle: 175/2 sqrt(3)

I have my solutions please check if they are correct.
 

Attachments

  • cir tri.jpg
    cir tri.jpg
    9.2 KB · Views: 341

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
43,021
970
That's not what I get.

Each side of triangle cuts the circle in a 360/3 = 120 degree arc.

Taking "r" as a radius of the circle, the two radii and one side of a triangle form a small triangle with 2 sides of length r and angle 120 degrees. By the cosine law, the side of the triangle has length s given by [itex]s^2= r^2+ r^2- 2r*r cos(120)= 2r^2+ 2r^2(-1/2)= 3r^2[/itex] so that [itex]s= r\sqrt{3}[/itex]. Dropping a perpendicular from the center of the circle to the side of the triangle gives a right triangle with hypotenuse of length r and one leg of length [itex](\sqrt{3}/2 )r[/itex]. By the Pythagorean theorem, The other leg has length given by [itex]x^2= r^2- (3/4)r^2= (1/4)r^2[/itex] so that [itex]x= r/2[/itex]. That is, each circle has radius exactly half the radius of the next larger circle.

If the outermost circle has radius R, then sum of the areas is the geometric series
[tex]\R^2+ \frac{1}{4}\pi R^2+ \frac{1}{8}\pi R^2+ \cdot\cdot\cdot[/tex]
[tex]= \pi R^2(1+ \frac{1}{4}+ \frac{1}{8}+ \cdot\cdot\cdot)[/tex].
 
  • #3
darkmagic
164
0
In my solution, first circle is 10 then the second is 5. after that, I used s = a/(1-r). a=25 since 5 will be squared and r=1/4. So my solution is A = pi[10^2 + 5^2 +25/(1-1/4). so my answer becomes 475/3 pi.
 

Suggested for: Circles and triangles

  • Last Post
Replies
4
Views
619
Replies
2
Views
476
  • Last Post
Replies
11
Views
956
  • Last Post
Replies
11
Views
228
  • Last Post
Replies
1
Views
1K
Replies
3
Views
72
Replies
27
Views
228
  • Last Post
Replies
5
Views
180
  • Last Post
Replies
7
Views
1K
Top