Finding the Area of an ArcDate: 1/23/96 at 13:55:33 From: Sweet Home Middle School Subject: Area of arc Dear Dr. Math, When you draw a circle and make a chord from one point to another, how would you find the area of that arc (formula)? TIA for your answer! Sincerely, Alyssa Ruffino, and Jillian Steinbroner Sweet Home Middle School, Amherst, N.Y. Date: 6/5/96 at 18:35:12 From: Doctor Charles Subject: Re: Area of arc As far as I know the shape formed is called a segment. There are two segments formed by this operation: the major segment and the minor segment. (The major one contains the centre of the circle.) To find the area of a minor segment think of it as a sector (slice of cake) minus a triangle with vertices at the centre of the circle and at the end points of the chord. Suppose the angle of the triangle at the centre of the circle is x degrees. Then the area of the sector is: Area of sector = pi * r^2 * x/360 (This formula comes from the formula of the area of a circle multiplied by the fraction of the circle that you are actually looking at.) The length of the chord is 2 * r * sin(x/2) but I shall call it 2L for now. (This comes from the definition of sin on the triangle formed by splitting the triangle which I have described above into two right angled triangles by drawing the perpendicular bisector of the chord.) Think of the chord as the base of the triangle. Then the height of the triangle is r * cos(x/2). Area of triangle = 1/2 * base * height = 1/2 * 2L * r * cos(x/2) = L * r * cos(x/2) So the area of the segment is the area of the sector minus the area of the triangle. Area of Segment = pi * r^2 * x/360 - L * r * cos(x/2) = pi * r^2 * x/360 - r^2 * sin(x/2) * cos (x/2) = r^2 * (pi * x/360 - (sin x)/2) Does this answer your query? -Doctor Charles, The Math Forum |
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