In square units, what is the area of the gray triangle?

The coordinates of the triangle are: (3, 0) (1, 3) and (6, 6).

 

To find the dimensions of this triangle, use the distance formula:

 

d = √((x₂ - x₁)² + (y₂ - y₁)²) 

Let's name the vertices for ease. A = (3, 0), B = (1, 3), and C = (6, 6)

 

AB = √((1 - 3)² + (3 - 0)²)
      = √((-2)² + 3²)

      = √13

BC = √((6 - 1)² + (6 - 3)²)

      = √(5² + 3²)
      = √34 

 

AC = √((6 - 3)² + (6 - 0)²)

      = √(3² + 6²)

      = 3√5

Now that we know the dimensions, we can use Heron's Formula for that. 

s = semi perimeter = (√13 + √34 + 3√5) ÷ 2 ≈ 8.07

A = √s(s-a)(s-b)(s-c)

   = √8.07(8.07 - √13)(8.07 - √34)(8.07 - 3√5)

   = 10.5 units², which is your answer :)

Base from  1,3  to  3,0      is      sqrt 13

Height fro   3,1 to  6,6      is       sqrt34

 

1/2 sqrt 34 * sqrt 13  =    1/2 (sqrt 442)   =  10.51 units^2