How can you calculate the chord within a circle?

Answer 1

It depends on which information is already known. You must know the radius and either the angle between the two radiithat end on the chord's endpoints (making a triangle with the center of the circle and the endpoints) or the distance from the chord to the center of the circle. There are two basic ways of finding the length of a chord in a circle. The first is if you know the radius and angle. This will be tough to explain in just words but I'll do my best.

Both processes work the same for these steps.

1. Draw a circle
2. Pinpoint (roughly) the center
3. Draw a chord (preferably in the bottom half of the circle)
4. Label the center of the circle point 'C'
5. Label the left endpoint 'A'

The first method continues like this:

1. Label the right endpoint 'B'
2. Connect C and A
3. Connect C and B
4. Label the center angle (which is ACB) 'θ'

If you know θ and CA/CB (the radius) then you multiply the radius by 2 and multiply this by the sin of half of θ. Much more simply: 2rsin(θ/2) If the angle is 60 degrees and the radius is 1 then: 60/2=30... sin(30)=0.5... 2*1=2...2*0.5=1 So the chord's length is 1 (Just a hint, if the angle is 60 degrees then the it will be an equilateral triangle, so the chord will always equal the radius at 60 degrees.

The second way would have continued like this:

1. Pinpoint (roughly) the center of the chord
2. Label that point B
3. Connect C and A
4. Connect C and B
5. Label the line CA 'b'
6. Label the line CB 'a'
7. Label the line AB 'c'

Here you need to find c and double it. The radius, which is b, forms the hypotenuse of a right triangle. And you should have been given the distance from the chord to the center, which is a. Pythagorean theorem: slightly altered but only because of labeling (you could rename them correctly by switching the label B with C and vice-versa and b with c and vice-versa). b2-a2=c2 To throw in some numbers, let's say the distance from the chord to the center was given as 3 and the radius is 5. plug these numbers in and you put the radius in for b and the distance in for a.

b(5)2-a(3)2=c2

52=25

32=9

25-9=c2

16=c2

the square root of 16 is 4

So the chord is twice that, 8.

Finally, a summery with the equations all written out.

With the radius and angle:

2rsin(θ/2)

With the radius and distance:

( (radius2-distance2)1/2 ) * 2

I really hope this helps. If you drew the circles it should have been fine. Good luck!