Area Definition & Proving Rectangle Area

[Avichal]

As far as i know a 1m by 1m square is defined to have 1 area unit. Following this definition or axiom how do you prove than area of rectangle is length multiplied by breadth?
Or is there another axiom that i don't know?

 

[Sourabh N]

I assume you have not studied integration. Here's an idea to obtain area of rectangle given knowledge of the area of a square.

Let the rectangle be of length l and breadth b.
1. Produce a square of edge l with one edge as the rectangle's one arm.
2. Produce a square of edge b with one edge as the rectangle's one arm.
3. Complete the square with edge (l+b). You will notice another rectangle in the square of the same size as the one you started with.
4. Area of (l+b)-edged square = area of rectangle + are of b-edged square + area of l-edged square + area of rectangle.

 

[Millennial]

You are getting stuck in the very definition of area. The area of a rectangle is defined to be the product of two sides, because it matches our intuition.

 

[Avichal]

Ok so area of rectangle is defined as such, but why product of length and breadth? I think there is some definition of area from which we can derive the area of all other things - if someone can just state the correct definition and derive the area of rectangle it would satisfy me.

 

[Mentallic]

Ok so area of rectangle is defined as such, but why product of length and breadth?

Because we've defined the unit of area as being a square with unit length, and it just so turns out that rectangles have the same property with squares which is that their sides are also perpendicular to each other.

I think there is some definition of area from which we can derive the area of all other things - if someone can just state the correct definition and derive the area of rectangle it would satisfy me.

I'm not sure what you're trying to ask.
I believe we could have also defined 1 unit area as being the area occupied by a circle with radius 1 unit (which we would could call 1 unit circled as opposed to pi units squared) but there are many reasons why we haven't done this. One of the most important reasons is that circles don't tessellate.

 

[Avichal]

Thank you - i am crystal clear now

 

[Byron Chen]

Ok so area of rectangle is defined as such, but why product of length and breadth? I think there is some definition of area from which we can derive the area of all other things - if someone can just state the correct definition and derive the area of rectangle it would satisfy me.

Let me try to answer this from a slightly different perspective. A line is one-dimensional, right? So you can give it a value (how long it is). Then move on to 2 dimensions, you probably know new space dimensions extends out perpendicular to the other existing dimensions. Area is 2 dimensional, so besides taking the value in the existing dimension, you must multiply it by the value of the new dimension.

Using the same intuition, we can move to higher dimensions. For volume (3D), you multiply the existing dimensions (area), by the new dimension, height. Even higher dimensions may be harder to visualise, but the mechanics remain the same.

 

[vincenttutor]

Basically, area is the amount of space inside a 2-d object.

I found this site interesting with clearly explained solved examples: http://www.swiftutors.com/math/cone.php