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johnb123
 Senior Boarder
Posts: 57 
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I am trying to find the area of a room which is roughly a quarter circle. One side is 10.2 m and meets the second side, 8.5 m in length, at a right angle. The third side is a smooth arc joining the other ends of the two sides. What is the area of this 'sector'? I've done an approximation but wonder if there is some exact formula for this type of calculation.
Chiffres
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Johnders
Expert Boarder
Posts: 86
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If the room is a quarter ellipse then the area is pi*10.2*8.5/4 square meters (about 68.1 square meters).
If the third side is a circular arc, then the answer depends on the radius of curvature of the circular arc. The smallest radius of curvature is sqrt(10.2^2+8.5^2)/2 meters (about 6.6 meters, when the arc is half a circle). This would give an area of 10.2*8.5/2 + pi*r^2/2 square meters (where r is the radius of curvature = about 6.6 m), or about 112.6 square meters. The largest radius of curvature is infinite, so the final side is a straight line segment. This would give an area of 10.2*8.5/2 = 43.35 square meters.
Carl G.
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jugherffere
 Expert Boarder
Posts: 84
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It depends on the details of the curved side. If it meets the two straight sides at right angles, and is an arc of an ellipse, then the area is 0.25 * pi * 8.5 * 10.2 square metres.
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paydayuscf
Senior Boarder
Posts: 79
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If you assume that the room is one-quarter of an oval, with semimajor axis 10.2 and semiminor axis 8.5, then choose your favorite way of finding the area of an ellipse (such as pi*a*b). In this case, 21.675*pi is the area
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Roger1955
Senior Boarder
Posts: 58
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I extrapolate from the recent thread about the volume of a balloon, and suggest that you place the room into a (large) bucket of water and measure the area empirically. Seems so obvious, doesn't it?
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Atraxani
Senior Boarder
Posts: 78
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While placing a room in a bucket of water is rather impractical, a related, and possibly practical method is this: Make certain that the floor is waterproof, and the walls (at least where they meet the floor), are, as well. Close the doors and calk around them (near the bottom; you don't have to make the entire door watertight). Now, fill up the room with exactly one centimeter of water, keeping track of what volume of liquid it takes to do so. The total number of cubic centimeters is also the number of square centimeters of floorspace.
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quest_marsman
Senior Boarder
Posts: 70
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Why not just look up the formula for the area of an ellipse, and take 1/4 of the area of a 20.4 x 17 ellipse?
Cheers!
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