# Area of rectangle and triangle derivation

I was wondering about the derivation for the area of a triangle and the area of a rectangle. Of course, we all know them to be $\dfrac{1}{2}bh$ and $bh$ respectively, but where is the derivation of these? If we derive the rectangle one then the triangle one obviously follows, but I don't recall ever seeing a derivation for the area of a rectangle. Is it like an axiom that we don't prove?

Area is defined as the number of smallest blocks of 1 unit that are required to fill in any 2-d geometry. Considering that, it is very easy to prove any area formula.

• How would you prove it? – user19405892 Feb 6 '16 at 14:42
• If the length of a rectangle is 'l', It covers 'l' blocks on the length axis. Similarly, It covers 'b' blocks on the breadth axis. The area will be the total number of blocks = l*b. – Win Vineeth Feb 6 '16 at 14:44
• That only seems to work for integers – Henry Feb 6 '16 at 14:48
• 1) It depends on what you choose as your smallest block. 2) You can consider parts of the box and add up all the boxes which are partially filled at the end. This is the basis for starting to work with areas. – Win Vineeth Feb 6 '16 at 14:51

I think that the area of a rectangle is the definition of multiplication. It is not an axiom, it just is multiplication. What you can do is derive the qualities of multiplication from this definition.

for example $ab=ba$ is true because a geometrical shape dose not change when moved.

Similarly you can prove that $$(a+b)c = ac+bc$$ $$ab=ac \Rightarrow b=c$$

etc.

One way to define area is to start with a $b \times h$ rectangle and require its area to be $bh$. See for example https://en.wikipedia.org/wiki/Area#Formal_definition which includes:

Every rectangle $R$ is in $M$. If the rectangle has length $h$ and breadth $k$ then $a(R) = hk$.

Then half the rectangle (a right-angled triangle) has area $\frac12 bh$.

Any acute triangle can be split into two right-angled triangles with a perpendicular from one vertex to the opposite side (giving the perpendicular height), while an obtuse triangle may be the difference between two right-angled triangles.

I have more on areas of triangles at http://www.se16.info/hgb/triangle.htm