# Find the radius of a circle from a chord

• Lizardjuice7
In summary, the equation to find the radius of a circle using the length of the chord and the distance from the midpoint of the chord to the edge of the circle is r = ((c^2)/4 + m^2)/(2m). This equation can be derived by creating a right triangle with the chord and the line from the midpoint to the opposite side of the circle, and using trigonometric relations to find the value of the radius.
Lizardjuice7
Hi,

Can someone please explain to me how you get the equation:

radius=($$\sqrt{c}$$+m2)/2m

c=length of chord

m=distance from midpoint of chord to edge of the circle

I would like to find the radius of the circle from only knowing those two quantities.

I have found the equation on various websites across the web (including http://www.math.utah.edu/~eyre/rsbfaq/physics.html), but I do not know how it was derived.

Thanks,

Lizardjuice7

From the link you gave they say the equation is r = ((c^2)/4 + m^2)/(2m), which is very good because that's the answer i just got from solving the problem. (EDIT: I think I see the problem, the link gives c^2/4 without parenthesis so you probably assumed c^(2/4) instead of (c^2)/4)

Draw your circle with a chord and a line from the midpoint of the chord to the opposite side of the circle. Chord is of length c and line bisecting the chord is of length m.

Now draw a line connecting the midpoint of the circle (which is on line m) to one of the corners of the chord. Because line m bisects the chord, this is a right triangle.

Consider the angle theta between the chord and the origin (the angle on your triangle which does not on the origin). Using normal trig relations:

sin(theta) = (m - r)/r
cos(theta) = c/(2*r)

Now just compute sin(theta)^2 + cos(theta)^2 = 1 and do the algebra!

thanks, that's a big help

## 1. How do you find the radius of a circle from a chord?

To find the radius of a circle from a chord, you can use the formula: radius = (chord length/2) / sin(angle between chord and radius)

## 2. Can the radius be found if only the chord length is given?

Yes, you will also need to know the angle between the chord and radius. This can be found by drawing a perpendicular line from the center of the circle to the midpoint of the chord. The angle between this line and the chord is the angle you need.

## 3. What if the chord is not perpendicular to the radius?

The formula still works, but you will need to find the angle between the chord and radius using trigonometry. You can use the inverse sine function to find this angle.

## 4. Is there another way to find the radius of a circle from a chord?

Yes, you can also use the Pythagorean theorem if you know the length of the chord and the distance from the center of the circle to the midpoint of the chord. The radius will be the square root of (half the chord length squared minus the distance squared).

## 5. What is the importance of knowing the radius of a circle from a chord?

Knowing the radius of a circle from a chord is important in many geometric and mathematical applications. It can be used to find the area, circumference, and other properties of a circle. It is also useful in engineering and construction when building circular structures.

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