There might be a better way, but here is one way.
Let the radius of circle A be a , the radius of circle B be b , and the radius of circle C be c .
From the given information, we can make these three equations:
a + b = 5 solve for a to get a = 5 - b
a + c = 6 solve for a to get a = 6 - c
b + c = 7 solve for b to get b = 7 - c
5 - b = 6 - c
Substitute 7 - c in for b .
5 - (7 - c) = 6 - c
Distribute the -1 to the terms in parenthesees.
5 - 7 + c = 6 - c
Rearrange
c + c = 6 - 5 + 7
2c = 8
c = 4 Use this value of c to find a and b .
a = 6 - c = 6 - 4 = 2
b = 7 - c = 7 - 4 = 3
area of circle A + area of circle B + area of circle C =
22π + 32π + 42π =
4π + 9π + 16π =
29π sq. units