# What is the area of a triangle with sides of length 4, 2, and 1?

##### 1 Answer
Jun 22, 2018

This is not possible to solve in real number solutions $\mathbb{R}$ because it is not possible to have a triangle with sides: $4 , 2 , 1$

$\frac{i \sqrt{105}}{4}$

#### Explanation:

To solve in the imaginary realm use Heron's formula:

$A r e a = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

where $s = \frac{a + b + c}{2}$ which is the $\frac{\text{perimeter}}{2}$

sides are $4 , 2 , 1$

$s = \frac{4 + 2 + 1}{2} = \frac{7}{2}$

$A = \sqrt{\frac{7}{2} \left(\frac{7}{2} - 4\right) \left(\frac{7}{2} - 2\right) \left(\frac{7}{2} - 1\right)} = \frac{i \sqrt{105}}{4}$