Find the orange area!

Redraw the figure above without affecting the orange region. We find the orange region is given by:

$\begin{aligned} A & = 30 \times 12 - \frac {30\times 10}2 - \frac {20\times 12}2 - 10 \times 2 \\ & = 360 - 150 - 120 - 20 = \boxed {70} \end{aligned}$

Define the points as shown and use the cross product formula for the area of a triangle. Calculate the areas of the two sub-triangles as shown, and combine them. The area comes out to $70$ .

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import math

x1 = 20.0
y1 = 12.0

x2 = 20.0
y2 = 10.0

x3 = 30.0
y3 = 10.0

A1 = 0.5*math.fabs(x1*y2 - y1*x2)
A2 = 0.5*math.fabs(x2*y3 - y2*x3)

A = A1 + A2

print A
# 70.0

Area of figure = $ar(ABC)$ + $ar(CBD)$ - $ar(green rectangle)$

Area of figure = $\frac{1}{2}*10*12 + \frac{1}{2}*2*30-10*2$

Area of figure = $\boxed{70}$