Adog is leashed to the corner of a house with an 18 foot leash. how much running area does the dog have?

If the dog were on a 15-ft leash tied to a stake in the ground 
out in the middle of a field, then he could cover every blade 
of grass in a 15-ft circle. The area of a circle is (pi R²), so 
he would have free range over (pi · 15²) square feet of area.

Sadly, he's not out in the middle of a field. He's tied to the corner 
of the house.  So he can't cover the whole circle, because the house 
blocks a quarter of the circle.  The dog only has access to

                       (3/4) of (pi R²)

                   =  (0.75) · π · (15²)

                   =  (0.75 · π · 225)  =  530 square feet  (rounded)