Hello, pyrosilver!

As Jhevon suggested, **make a sketch**!

Find the total grazing area of a goat.

The animal is tied to the notheast corner of a 40' by 40' barn by an 80' rope.

The south side of the barn is extended by a fence.

Assume that there is grass everywhere except inside the barn. Code:

* * *
*:::::|:::::*
*:::::::|:::::::*
*::::::::|80::::::*
:::::::::|:::::::::
*:::::::::|::::80:::*
*----*----*---------*
*::::| 40 |:::::::::*
*::| |::::::::*
- - - -*- - * - - * - - - -

The upper region is a semicircle with radius 80 feet.

In the lower-left, there is a quarter-circle with radius 40 feet.

In the lower-right, there is a strange-shaped piece. Code:

A * - - - - - - - - * D
| * 80 *
40 | *
| 30° * *
B * - - - - - - - * C
40√3

Right triangle $\displaystyle ABC$ has side 40 and hypotenuse 80.

From elementary Trig, $\displaystyle \angle ACB = 30^o = \angle DAC$

. . and hence: $\displaystyle BC = 40\sqrt{3}$

$\displaystyle \Delta ABC$ has base $\displaystyle 40\sqrt{3}$ and height $\displaystyle 40.$

Sector $\displaystyle ADC$ has $\displaystyle \frac{1}{12}$ of the area of a circle with radius 80.

I assume you can find those separate areas ... and add them.