Square Yards, Area

Check your units. The diameter of the lake and the walk is in feet.

Rap

You are using the wrong formula for the area of the walkway.

Area of ring = pi ( (R+r)^2 - R^2)

Also as Rap mentioned, there are 9 sq ft per sq yd.

Pi(R^2 - r^2)
Pi(120^2 - 3^2)
Pi(14400 - 9)
Pi(14391)
3.1416(14391)
45210.7656 = 45211 square feet

452111/9 = 5023.444= 5023 square yards

??

Read the formula again.

Pi ( 123^2 - 120^2)

Pi (123^2 - 120^2)
Pi (15129 - 14400)
Pi (729)
3.1416 (729)
2290.2264 = 2290 square feet
2290/9 = 254.444 = 254 square yards

How do I proceed from here?

First, I made a mistake and we need to correct it. The radius of the lake is indeed 120, but the radius of the outer circle is not 123, it is 126. The walkway is 6 feet in width, you don't need to half it, so use

Pi (126^2 - 120^2) for the area but otherwise your answer is correct.

For the second part where the walkway is doubled in width, do the same thing but use 132 for the outer radius.

Pi (132 ^2 - 120 ^2)
Pi (17424 - 14400)
Pi (3024)
3.1416 (3024)
9500.1984 = 9500 square feet
9500/9 = 1055.555 = 1056 square yards