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 June 24th, 2011, 08:06 PM #1 Newbie   Joined: Jun 2011 Posts: 21 Thanks: 0 Finding Sides and area of trapezoid around two circles. Find Sides and area of trapezoid around two circles with radius=1 and Radius=3. Here is the exact picture give by teacher (not drawn to scale) That was the only info given. Any guidance or suggestions on how to solve?
 June 24th, 2011, 09:57 PM #2 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Assume the circles are externally tangent and the trapezoid is isosceles. The sides are $\frac{2\sqrt{3}}{3}\text{, }\frac{16\sqrt{3}}{3}\text{, }\frac{16\sqrt{3}}{3}$ and $6\sqrt{3}\,>\,\frac{2\sqrt{3}}{3}\,\text{.}$ The area is $\frac{80\sqrt{3}}{3}\,\text{.}$ I'm 99% certain that my answers are correct.
 June 25th, 2011, 06:58 AM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Re: Finding Sides and area of trapezoid around two circles. Orienting the trapezoid so the r = 3 circle is at the bottom and assuming the centers of the circles are in a line perpendicular to the base of the trapezoid, the angle in the bottom two corners is arctan(2) and the height of the trapezoid is 8. Thanks from Dino89
 June 25th, 2011, 08:46 AM #4 Newbie   Joined: Jun 2011 Posts: 21 Thanks: 0 Re: Finding Sides and area of trapezoid around two circles. Where did you get the number 2 from?
 June 25th, 2011, 12:40 PM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond Re: Finding Sides and area of trapezoid around two circles. From a mistake, sorry about that.
 June 25th, 2011, 02:40 PM #6 Newbie   Joined: Jun 2011 Posts: 21 Thanks: 0 Re: Finding Sides and area of trapezoid around two circles. im still confused. how would i find the area of the trapezoid though?
 June 25th, 2011, 03:22 PM #7 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Height 8 with bases $\frac{2\sqrt{3}}{3}$ and $6\sqrt{3}$ therefore area $\frac{h(b_1\,+\,b_2)}{2}\,=\,\frac{8(\frac{2\sqrt{ 3}}{3}\,+\,6\sqrt{3})}{2}\,=\,4(\frac{2\sqrt{3}}{3 }\,+\,6\sqrt{3})\,=\,\frac{80\sqrt{3}}{3}\,\text{. }$
June 25th, 2011, 04:18 PM   #8
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Re: Finding Sides and area of trapezoid around two circles.

Why can't the picture be drawn like this? Then the area is 30+16 sqrt(3)
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 June 25th, 2011, 06:42 PM #9 Newbie   Joined: Jun 2011 Posts: 21 Thanks: 0 Re: Finding Sides and area of trapezoid around two circles. How did you find the length of the sides?
June 25th, 2011, 07:12 PM   #10
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Re: Finding Sides and area of trapezoid around two circles.

Quote:
 Originally Posted by Spaghett How did you find the length of the sides?
From the centers of the circles, draw all the radii to the points of tangency. With r=1 and R=3 you should get lots of 30-60-90 triangles (probably similar to the method that johnny used). The height of the trapezoid (the way I drew it) is $3 + 1 + 2\sqrt{3}$.

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