Formula for the Length of a ChordDate: Wed, 3 Jul 1996 10:11:02 -0400 (EDT) From: Anonymous Subject: Length of chord I'm trying to find a formula that will give me either the length of a chord when I know the circle radius and the distance from center to chordline OR the area of the resulting circle segment. Date: Wed, 3 Jul 1996 12:07:22 -0400 (EDT) From: Dr. Anthony Subject: Re: Length of chord The first of these is easy, since you can use Pythagoras to find the length of half the chord (1/2*c) from a right-angled triangle. The hypotenuse is equal to the radius (= r) of the circle, and the perpendicular distance (= d) from the centre of the circle to the chord is also given. So we have: {(1/2)*c}^2 = r^2 - d^2 (1/4)*c^2 = r^2 - d^2 c = 2*sqrt(r^2-d^2) The second is a little more complicated. We find the area of the segment by subtracting the area of triangle OAB from the area of the sector OAB of the circle, where O is the centre of the circle, and AB the chord. The angle AOB can be calculated from cos(1/2*AOB) = d/r, so 1/2*AOB = arccos(d/r) Therefore AOB = 2*arccos(d/r) (Give AOB in radians) Area of sector = (1/2)(AOB)r^2 (AOB in radians) Area of triangle AOB = (1/2)sin(AOB)*r^2 Area of segment = r^2*[arccos(d/r) - (1/2)sin(2*arccos(d/r))] -Doctor Anthony, The Math Forum |
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