Area outside equilateral triangle available for grazing by the goat.

Area of circular sector
$A = \dfrac{\pi r^2 \theta_{deg}}{360^\circ}$
 

Area A₁: central angle = 300°, radius = 15 ft.
$A_1 = \dfrac{\pi(15^2)(300^\circ)}{360^\circ}$

$A_1 = \frac{375}{2}\pi ~ \text{ft.}^2$
 

 

Area A₂: central angle = 120°, radius = 5 ft.
$A_2 = \dfrac{\pi(5^2)(120^\circ)}{360^\circ}$

$A_2 = \frac{25}{3}\pi ~ \text{ft.}^2$
 

Area available for grazing by the goat
$A = \frac{375}{2}\pi + \frac{25}{3}\pi$

$A = \frac{1225}{6}\pi ~ \text{ft.}^2$