Perimeter of 1000mDate: 07/13/99 at 14:59:31 From: Louise Macdonald Subject: A shape with the maximum area and a perimeter of 1000m We have to find a shape with a perimeter of 1000m and the maximum area. I have tried to work this out but need to know if a circle would have the biggest area. I have already looked at rectangles, triangles, and polygons and have noticed that the equilateral shape had the maximum area of each group of shapes. Thank you. Date: 07/13/99 at 15:09:48 From: Doctor Anthony Subject: Re: A shape with the maximum area and a perimeter of 1000m Consider various regular polygons with n = number of sides. You will have a constant perimeter (= 1000m) and you want an expression for the area in terms of n. When n = 3 you have an equilateral triangle with each side 333.33 metres. area = (1/2)333.33^2 sin(60) = 48112.52 m^2 For a square of side 250 m area = 62500 m^2 From here you require a general formula. If a polygon has n sides, the angle subtended by any side at the centre is 360/n. Suppose the polygon is inscribed in a circle of radius r; then the area of any triangle formed by radii from the centre to the edges of a side of the polygon is (1/2)r^2.sin(360/n) and the area of the whole polygon is (n/2)r^2.sin(360/n) where the perimeter is given by 2n.r.sin(180/n) = 1000 n.r.sin(180/n) = 500 So you have two formulae to work with: r = 500/[n.sin(180/n)] .....(1) and area A = (n/2)r^2.sin(360/n) .......(2) Check with n = 3 r = 500/[3.sin(60)] = 192.4500897 A = (3/2)(192.4500897)^2 sin(120) = 48112.52 and this agrees with the value we found earlier. Combining the two formulae we get the area of a regular polygon of n sides having a perimeter of 1000m: (500)^2 A = ----------------- n.tan(pi/n) which can be changed to (500^2/n).cot(pi/n) = (500^2/n).tan(pi/2 - pi/n) or if you want to express it in degrees A = (500^2/n).tan(90 - 180/n) Plot a graph of A against n and show that A increases with an increase in n, but at a rate that decreases as n increases. The obvious limit is when you have n -> infinity, in which case you have a circle. Find the area of the circle. This will be the limit toward which A is heading as n is increased. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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