What is the largest possible area of a rhombus constructed from an 11 by 8-1/2 inch paper?

Question

PLEASE HELP!

Answer 1

The largest rhombus will be the size of the largest square you can remove, namely 8½ in x 8½ in = 72¼ in²

Its interior acute angle can vary from ~73°37' (arctan 8½/2½) to 90° and its area will not change as Area = base x height and the height is fixed at 8½ in.

But the perimeter will vary from 34in (the square) to 34.72in (the most inclined rhombus)

i. The most inclined rhombus

<------------ 8½ in ------------->
_______________ ___________
|\................ ............. ............. \............|
|..\.............. ............. ............. ..\..........|
|....\............ ............. ............. ....\........|
|......\.......... ............. ............. ......\......| 8½ in
|........\........ ............. ............. ........\....|
|..........\...... ............. ............. ..........\..|
|______\________ __________\|
<---------------- 11 in ----------------->

ii. The square
............ <--------- 8½ in --------->
______.______ __________
|..............|............ ......................|
|..............|............ ......................|
|..............|............ ......................|
|..............|............ ......................| 8½ in
|..............|............ ......................|
|______|_____ _________.|
<-------------11 in ------------->


Sorry Stephen but you are incorrect as the area of a rhombus is length of base x height NOT product of side lengths

And Jeff a square IS most definitely a rhombus ... just a special one. It has every single property a rhombus has .. and more. Squares are subsets of rhombuses just as rhombuses are subsets of parallelograms.

Furthermore the definition of a rhombus does NOT include the statement 'No right angles'. In fact a rhombus is defined as a parallelogram with adjacent sides equal.

You cannot exclude squares from the family of rhombuses just because they have a right angle ... that is discriminatory!!

Answer 2

A rhomus is similar to a slanted square. So, I would personally go with an 8x8 rhombus, 64 inches. you could get a slightly larger size, but it would be difficult, as to achieve a rhombus, you need the following things:

2 sets of 2 parallel lines
4 lines of equal length
angles in opposite corners of equal angle
angles at the end of any one line add up to 180 degrees
no right angles

if all requirements are met EXCEPT the last one, you no longer have a rhombus, but a square. There is some debate as to whether or not a rhombus is a square, but I would suggest you err on the side of caution and meet ALL requirements

Answer 3

The largest rhombus will have its opposite vertices on opposite corners of the paper - call these A and C, with the other two vertices on the two long sides (B and D).
Let length of AB = x. Then we get a right angled triangle with sides 8.5, (11-x) and x.
So 8.5^2 + (11-x)^2 = x^2. Solving that equation gives x = 773/88.
Thus the area = 8.5x = 74.66, which is bigger than the above answer :)