Question

How do you use Heron's formula to find the area of a triangle with sides of lengths `1`, `1`, and `2`?

Answer 1

Heron's formula for finding area of the triangle is given by
`Area=sqrt(s(s-a)(s-b)(s-c))`

Where `s` is the semi perimeter and is defined as
`s=(a+b+c)/2`

and `a, b, c` are the lengths of the three sides of the triangle.

Here let `a=1, b=1` and `c=2`

`implies s=(1+1+2)/2=4/2=2`

`implies s=2`

`implies s-a=2-1=1, s-b=2-1=1 and s-c=2-2=0`
`implies s-a=1, s-b=1 and s-c=0`

`implies Area=sqrt(2*1*1*0)=sqrt0=0` square units

`implies Area=0` square units

Why is are 0?

The area is 0,because there exists no triangle with the given measurements the given measurements represent a line and a line has no area.

In any triangle the sum of any two sides must be greater than the third side.

If `a,b and c` are three sides then
`a+b>c`
`b+c>a`
`c+a>b`

Here `a=1, b=1` and `c=2`

`implies b+c=1+2=3>a` (Verified)
`implies c+a=2+1=3>b` (Verified)
`implies a+b=1+1=2cancel>c` (Not Verified)

Since, the property of triangle is not verified therefore, there exists no such triangle.