My Math Forum segment of a circle problems

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October 1st, 2012, 11:39 AM   #1
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segment of a circle problems

Hi I am have a bit of difficulty working out the two diagrams that I have attached to this post.

The first picture I am a little confused by the dotted line, see I work out the area of the triangle by:$\frac {1}{2}\ r^2\sin \theta$ I then get slightly confused do I work out the sector? I don't really know because I not to sure if it's in a circle or semi circle and if in a semi circle do I: $\frac{\theta}{180}\pi\ r^2$.

Here is the second picture with the question: An Australian sound board is made out of a rectangular piece of wood 30cm by 4cm. It consists of a rectangle with a segment of a circle at each end.

a) find the perimeter of the board: For this part, I first started by using the tan rule: $\frac {1.5}{2}\=1.3333333333$ $atan\= 53.1$

I then $\times 2= 106.2$. Once I had done this I then did, I work out area of segment: $\frac{\ 106.2}{360}\pi\ 2.5^2$ - $\frac {1}{2}\ sin 106.2$. After this, I take the answer away from for the area of a rectangle in this case: 30 x 4 = 120.

b) Find the perimeter: I have not answered this due to the fact I got the first answer wrong.

I would appreciate, if someone could put me in the right direction, it would be very helpful.
Attached Images
 austrailian board.jpg (31.3 KB, 147 views) segment circle.jpg (26.8 KB, 147 views)

 October 1st, 2012, 12:33 PM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: segment of a circle problems Can you post the questions exactly as they are in the book? It is not clear what you mean by "sector" and "segment". If need be, please clarify.
 October 1st, 2012, 01:03 PM #3 Senior Member   Joined: Sep 2012 Posts: 201 Thanks: 1 Re: segment of a circle problems The first question is find the area of the shaded segment, which is part shaded blue. The Australian sound board one is exactly how written. It tells me to use Pythagoras theorem to find the radius, which I did.
 October 1st, 2012, 01:35 PM #4 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: segment of a circle problems Repost the second diagram with all relevant measurements included. As it stands I'm getting a perimeter of about 69.3 for part a). There's a point of confusion here: for part b) they ask "find the perimeter". Perimeter of what? Also, shouldn't the tangent ratio be 2/1.5?
 October 1st, 2012, 01:40 PM #5 Senior Member   Joined: Sep 2012 Posts: 201 Thanks: 1 Re: segment of a circle problems I will redraw the diagram and post.
 October 1st, 2012, 02:04 PM #6 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: segment of a circle problems First problem: The formula for the area of the circular sector is $\pi r^2\,\cdot\,\frac{\theta}{360}$ where $r$ is the radius and $\theta$ is the angle at the origin of the circle. So we have $\pi(27)^2\,\cdot\,\frac{32}{360}\,=\,\frac{2916\pi }{45}$ Now, to calculate the area of the triangle formed by the radii of the circle and the flat part of the segment, we use $\frac{bh}{2}$ where $b$ is the base of the triangle and $h$ is the height: $\frac{b}{2}\,=\,27\sin{16}$ $h\,=\,\sqrt{27^2\,-\,(27\sin{16})^2}$ So, for the area of the segment we have $\frac{2916\pi}{45}\,-\,27\sin{16}\,\cdot\,\sqrt{27^2\,-\,(27\sin{16})^2}\,\approx\,10.4196\,\text{cm.}$
October 1st, 2012, 02:24 PM   #7
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Re: segment of a circle problems

Thanks for the reply for the first diagram, I sort of see where you are coming from; probably understand better in the morning, it's been a long day. I have attached an exact copy of the second diagram. Hope this helps.
Attached Images
 mathforumpic..jpg (38.0 KB, 127 views)

October 1st, 2012, 05:13 PM   #8
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Quote:
 Originally Posted by taylor_1989_2012 a) find the perimeter of the board
Did you mean "perimeter"? You proceeded to calculate areas. Also is 30cm the entire length? This differs in your two diagrams.

 October 2nd, 2012, 12:03 AM #9 Senior Member   Joined: Sep 2012 Posts: 201 Thanks: 1 Re: segment of a circle problems 30 cm of whole board, sorry the first diagram was done in a rush. I have to calculate area of board and perimeter of the board.
 October 2nd, 2012, 12:52 AM #10 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond Re: segment of a circle problems The circumference of a circle is pi times the diameter, so the length of the two rounded ends of the board is 4?. The straight edges of the board are of length 30 - 4 = 26, so the perimeter of the board is 4? + 52 ? 64.5664 cm. The area of a circle is pi times the square of the radius and the area of a rectangle is length times width, so the area is 4? + 26 × 4 ? 116.5664 cm.

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