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Thread: Finding the radius with just the chord length and central angle :)

  1. #1
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    Finding the radius with just the chord length and central angle :)

    Hello everyone,

    I'm an product design that has come across a slight problem due to poor maths ability. I would like to find out the radius of a circle segment whilst only knowing the chord length (150) and the central angle (18).

    Please, if someone could help me solve this I would be extremely grateful. It's very frustrating since i've not done any trigonometry in years and it is vital for me to progress in my project.

    Thank you for your time

    Belinda
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Belinda View Post
    Hello everyone,

    I'm an product design that has come across a slight problem due to poor maths ability. I would like to find out the radius of a circle segment whilst only knowing the chord length (150) and the central angle (18).

    Please, if someone could help me solve this I would be extremely grateful. It's very frustrating since i've not done any trigonometry in years and it is vital for me to progress in my project.

    Thank you for your time

    Belinda
    Draw a diagram (see attachment).

    From this we see that sin(9)=75/r, or r=75/sin(9). Now the sin of 9 degrees is ~= 0.1564,
    so r ~= 479.4

    RonL
    Attached Thumbnails Attached Thumbnails Finding the radius with just the chord length and central angle :)-gash.jpg  
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  3. #3
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    Quote Originally Posted by Belinda View Post
    Hello everyone,

    I'm an product design that has come across a slight problem due to poor maths ability. I would like to find out the radius of a circle segment whilst only knowing the chord length (150) and the central angle (18).

    Please, if someone could help me solve this I would be extremely grateful. It's very frustrating since i've not done any trigonometry in years and it is vital for me to progress in my project.

    Thank you for your time

    Belinda
    Just in case you come across another problem like this.

    radius=(half the chord length)/(sin of half of the angle)
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    Thank you both for replying
    I realised after posting that the simplest way to find out was to draw a diagram like captainblack suggested. Thank you for the equation quick.

    I got 478 on my diagram, I thought I'd be way off.
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  5. #5
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    Re: Finding the radius with just the chord length and central angle :)

    Thank you for asking the question Belinda. I am cutting concrete pavers into a 90deg arc around a tree in the corner of a garden, and wanted to cut the angle off each side of each block to have an even gap between them. I wanted to know the radius so I would know were to place the front of each block so that the angle I had just cut maintained even gaps around the arc. Using your question and the knowledgeable answers I was able to calculate the angle to cut the blocks and the distance from the corner to start the placement.
    Sorted!
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  6. #6
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    Re: Finding the radius with just the chord length and central angle :)

    Another way of looking at it is that, since the two radii are equal length, you have an isosceles triangle with the base of length 150 and vertex angle 18 degrees. By the cosine law, $\displaystyle 150^2= a^2+ a^2- 2a(a)cos(18)= 2a^2(1- cos(18))$.

    Or you could use the sine law. Since the two sides are of equal length the two base angles are also the same. The vertex angle is 18 degrees so the two base angles are $\displaystyle \frac{180- 18}{2}= 90- 9= 81$ degrees. By the sine law, $\displaystyle \frac{a}{sin(81)}= \frac{a}{sin(18)}= \frac{150}{sin(18)}$. So $\displaystyle a= 150\frac{sin(18)}{sin(81)}$
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  7. #7
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    Re: Finding the radius with just the chord length and central angle :)

    Good to know. Incidentally, I calculated that at a point 1224mm out from the corner (centre of the circle) I would need to place the outside edge of each of 10 blocks with a (cord) side length of 193mm. Each block has a long side or 233mm corresponding to the radius, so if each block occupies a 9 degree arc, then each side of each block occupies 4.5 degrees, so using the sin rule I calculated the start of my side cuts to be 18.3mm from the inner end of the block tapering out to nothing at the outside face.
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