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What properties maximize area?

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by clawhammer » Sat Dec 04, 2010 5:33 am
Guys, I need the concepts of when an area is maximized:

For example, when does a isosceles triangle have greatest possible area? - When it's a right triangle.

Similarly:

1. Any other similar logic for equilateral triangles?
2. Other triangles?
3. Any specific scenario maximizes area for any specific type of Quadrilaterals?

Please let me know if isosceles triangle is the only one where this type of condition holds, or share your knowledge for this concept.
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by Rahul@gurome » Sat Dec 04, 2010 5:53 am
clawhammer wrote:Guys, I need the concepts of when an area is maximized:

For example, when does a isosceles triangle have greatest possible area? - When it's a right triangle.

Similarly:

1. Any other similar logic for equilateral triangles?
2. Other triangles?
3. Any specific scenario maximizes area for any specific type of Quadrilaterals?

Please let me know if isosceles triangle is the only one where this type of condition holds, or share your knowledge for this concept.
The answer to this question depends upon what properties of the triangle/quadrilateral is fixed. You have to fix some properties of the triangle/quadrilateral to maximize its area. Otherwise there is no maximum limit for the area! It can go on increasing with increasing length of sides.

But for isosceles triangle, when the area is maximum a particular condition holds. If you are asking for that particular condition, say an isosceles triangle as shown in the figure,
Image
Then the are of the triangle is given by,
Image

Now for maximum area there is a relation between a and b, which is a = (√2)b. Try to see how it comes. Otherwise I'll help.

For equilateral triangles or square or rectangle, the area is only dependent on the length of its sides. Thus if you have fixed the length of the sides, their area are also fixed.

In general, find the expression for area, then maximize it depending upon the constraints given.
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by TOPGMAT » Sun Dec 05, 2010 12:43 am
Hi Rahul,
Can you pls explain how ?


product xy is maximum when x=y given x+y=const.

so if 1/2a=(b^2-1/4a^2)^1/2
==> we get a = 2^1/2 b.

correct ?
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