maths problems

[randy-andy]

can u help me pleese. if you find the area of lots of irregular and regular rectangles and triangles, the ones with the biggest area are the regular ones. WHY ???. i need to know why they are bigger. thank you

[teacher1]

Area of triangles is measured by the base length time the height - In the case of irregular and regular triangles of the same base length the only difference is the height. Therefore the higher the apex (top) the more the area.
This also works with rectangles with the same base length.

Hope this makes sense

[teacher1]

That should be TIMES the height - not time. sorry.

[teacher1]

Actually thinking about it, I'm not sure I'm right.

[bridgenut]

Thought the area of a triangle was half the base times the perpendicual height ?

[teacher1]

You are perfectly correct bridgenut - fools rush in where ............. That's me !!!!!!!

[teacher1]

Please forget my ramblings above randy-andy.

[gen2]

randy-andy's statement as written is false. A very large irregular polygon can easily have a greater area than a very small but regular polygon. Maybe what randy-andy intended to include was a proviso that all the polygons have the same fixed perimeter.

[TeeGee]

You get the same answer if you take half the perpendicular height times the base, as the formula is
a = ( B * H ) / 2

So, we all win !

[Aquagility]

Is this because they more nearly approximate to a perfect circle, which has the greatest area for a given perimeter?

[mohill]

gen2 and aquagility make the two key points. You need to be speaking of shapes where the length of the perimeter is fixed. The largest area is covered by the 'perfect' shape ie a circle. When you move into areas with straight line sides, you have polygons, such as octagons or pentagons or squares or triangles and so on. "Regular' polygons (ie all sides are equal and all angles are equal) have a larger area for a fixed perimeter that irregular polygons with the same number of sides. Also the more sides the polygon has for the fixed perimeter then the greater will be its area
That means that a square (which is regular) will have a greater area than a rectangle, if both have the same perimeter. It also means that a regular pentagon will have a greater area than a square, because it has more sides.

You asked WHY? It can be proved quite easily using elementary calculus, but that may or may not be beyond your background.
Cheers

[Aquagility]

An easy way of thinking about it, without the use of calculus, is this:
As mohill says, a square has a greater area than a rectangle (for the same perimeter). Why?. Well, imagine a very long, thin rectangle. The longer and thinner it gets, the less area it has until, finally, when the two long sides meet it has no area at all. it has become a straight line.
I hope this helps!

[randy-andy]

im sorry if i wasnt clear but they ALL HAVE THE SAME PERIMETER.