Triangle With No Area

We know the formula for the area of triangle is $\dfrac{1}{2}ab\sin C$ . For the area to be zero we need, by the Zero Product Property , $a=0$ or $b=0$ or $\sin C=0$ . We know that the sides of a triangle are always positive. so $a,b\not\in\{0\}^2$ . We are left with $\sin C=0 ~\Rightarrow~ C=0, ~\pi~ (\text{range of triangle angles as the sum is}~\pi)$ As a triangle cannot have 0° or 180° , we see that none of our solutions are possible, and conclude that a triangle always has a positive area.

The triangle inequality says that, for a triangle with sides a, b, and c, a+b > c.

If we let the zero side be a, then b = c; however, by the inequality we have b > c.

Thus, we have a contradiction, and our initial assumption is false.

It is not possible that a bounded polygon doesn't enclosed region in plane it is possible when figure is not bounded polygon

The two legs of a triangle cannot both equal 0 in length, so the hypotenuse cannot equal 0. Therefore, no two sides can equal 0. A=0.5bh. Also, if a figure had zero area, wouldn't it just be a point? To extend on the above, no side or angle can have a negative length or measure, so we cannot have two opposites in a triangle. (i.e.: a whole number and its opposite, such as 1 and -1.). Consequently, no combination of sides or angles could cancel each other out. link text

If The Triangle Can Have Zero Area Than It Is A Point.