Area of a Circle SegmentDate: 04/18/99 at 16:21:19 From: Blair Subject: Area of a segment What I need help on is figuring out the formula of steps for how to figure out the area of a segment. For example, take a circle with an angle of 120 degrees. Where the two radii meet with the circle, a line is drawn to connect them. The area of the segment is the area from that line connecting the two radii, to the outer edge of the circle. How would you figure this out? Date: 04/19/99 at 12:36:20 From: Doctor Peterson Subject: Re: Area of a segment Hi, Blair. To find this area in general, you need to use trigonometric functions. Here's a place to find the formula in our FAQ (scroll down to Segment of a Circle): http://mathforum.org/dr.math/faq/formulas/faq.circle.html In your case, since you know the angle, and your angle is one of the easy ones to work with, you can do without this. Here's how you can come up with a formula. You want the area of a segment of a circle, between a chord and its arc: *** * * * * * * * * * +---------*---- * r / t \ * | * / \ * |h *---------------* ----- *###########* *** You know the angle t and the radius r. Given that t is 120 degrees, you can find h, the altitude of the triangle between the two radii. (I'll leave that part for you to do.) You can think of this area as a sector (bounded by the arc and two radii) minus a triangle (bounded by the radii and the chord): sector: *** * * * * * * * * * + * * /######\ * * /############\ * *###############* *###########* *** triangle: *** * * * * * * * * * + *---- + * /######\ * | | \ r * /############\ * |h h| \ *---------------*------ +---------+ * * b *** The triangle has height h, and base 2*b, where b is the other side of the right triangle with hypotenuse r and side h). So the area is A = 2*b * h / 2 = h * sqrt(r^2 - h^2) The angle at the top of the sector is t = 120 degrees. Since the area of a sector is proportional to the angle, t A = ----- * pi * r^2 360 pi * r^2 * t = ------------ 360 The area of the segment is the difference of these, or pi * r^2 * t A = ------------ - h * sqrt(r^2 - h^2) 360 - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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