Maths Conversion

[BlueToffee]

OK, brain dead today.

Can anybody please help me convert a perimeter of 180 metres into square metres, it's a plot of land.

Thanks in advance.

[jim360]

Not an easy shape to work out at all, since it's not even obviously a cyclic quadrilateral or anything. There are formulas that exist, but it's hard to find a useable one.

[Peter Pedant]

trapezoidal - someone else can do that

[Zacs-Master]

I make it 180 sq m. I drew it in Autocad which works out the area for you, so I'm not sure what your OP means.

[Zacs-Master]

not true jim. All you have to do is draw 3 sides. draw a circle 63m from the end of the last line. Go back to the start of the first line and rotate it to meet the arc of the circle. join the points and VIOLA!

[jim360]

The main point is that there is no "this is the perimeter, here's the area" formula -- because different shapes of the same area have different perimeters, and different shapes of the same perimeter have different areas. This should be intuitively obvious, by considering rectangles of a fixed perimeter 2*(a+b) that get thinner and thinner. Choosing, say, a=b=1 gives our starting perimeter of 2*(1+1) = 4, and an area of 1*1 = 1. Choosing now a = 3/2 and b=1/2 still gives a perimeter of 2*(3/2+1/2) = 4 but an area now of (3/2)*(1/2) = 3/4, which is less than 1.

[Zacs-Master]

Funny how it worked out exactly then.

[jim360]

Not sure what you are saying, Zacs, since a given quadrilateral is usually not cyclic, and sometimes you can't even make it cyclic either. And even two quadrilaterals can have the same side lengths but a different area depending on how they are constructed. So a square having sides of length 2 has an area of 4; but a rhombus with the same side length has an area less than 4. The shape matters!

[EcclesCake]

How about getting a sheet of graph paper. Drawing the plot and then counting the squares?

[Zacs-Master]

Jim
http://i58.tinypic.com/2hnmc1h.jpg

[jim360]

That's not a cyclic quadrilateral though... a cyclic quadrilateral has a circle that touches all four corners.

[Zacs-Master]

So how come it works. I have no idea what a cyclic quad is. I just drew it.

[jim360]

I'm not sure it's clear that you have the unique shape with those side lengths. As I mentioned in the case of rhombus and square, there is a literally infinite set of shapes with the same four side lengths. For a full construction you'll need to draw:

one side of length 36

two circles from either end, of radii 46 and 63 (and this assumes that Bluetoffee has read his side lengths in an order about the quadrilateral);

a remaining side of length 35 connecting two points on those circles you drew.

If, when drawn that way, there is a single possibility, then you've found the unique shape, and you'll be able to calculate its area. But I don't think you've shown that you've found the unique shape, and each shape will have a different area, so the problem is yet to be convincingly solved. And I'm amazed that the area comes out as 180 sq m, since the way you've drawn it a decent lower bound would be 35m*46m = 1,610 m^2.

[Zacs-Master]

Strange. AutoCAD calculates it as 179.995 sq m actually. Maybe I just stumbled upon the exact shape (you can work out the odds of that if you like ;-))

[Zacs-Master]

I just tried drawing some alternatives and they don't add up to the 180. How strange. I'm off to have a bet on a rank outsider......

[Prudie]

Amusing how often maths brings out the arguments on here. It's a simple fact that you need to know the shape before you can work out the area.
I'd thought this piece of land exists already so making a shape to fit the exotic formulae is pointless - and assuming it's 4 straight sides pi has nothing to do with it.

[sandyRoe]

If somebody could calculate the diagonal would trig enable them to find the area of the two triangles and add them together to give the total?

[jim360]

Yes -- but finding the diagonal is non-trivial, because in general there are many solutions. I've been busy trying to solve this problem and loosely speaking you get a quartic equation with one free variable, which translates into a whole range of solutions, each of which will then have a different area.

[gness]

I think Blue has gone off to find a different plot of land....☺

[jim360]

There are some seriously interesting graphs coming out of this problem!

[Zacs-Master]

LOL at Gness. All I know is the square on the hippopotamuse is equal to the sum of the square on BBC 1 and Channel 4.