Given:
- A circular sheet of area A.
- Two equal circular regions are cut off from the sheet.
- We need to find the remaining area of the sheet.

Finding the area of the circular sheet:
- The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
- Since the sheet is circular, we can assume that the entire area is accounted for by the circle.
- Therefore, the area of the circular sheet is A.

Finding the area of each circular region:
- Since the two circular regions are equal, let's denote the area of each region as X.
- The total area of the two regions is 2X.

Finding the remaining area of the sheet:
- The remaining area of the sheet is the difference between the area of the sheet and the area of the two circular regions.
- Remaining area = A - 2X

Finding the greatest possible area for each circular region:
- To find the greatest possible area for each circular region, we need to maximize the radius of each region.
- The radius of each region can be maximized when the two regions are touching each other at a single point, dividing the sheet into two halves.
- In this configuration, the diameter of each region is equal to the diameter of the sheet.
- Therefore, the radius of each region is half the radius of the sheet.
- Let's denote the radius of the sheet as R, then the radius of each region is R/2.

Calculating the area of each circular region:
- The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.
- The area of each circular region is X = π(R/2)^2 = πR^2/4.

Calculating the remaining area of the sheet:
- Remaining area = A - 2X = A - 2(πR^2/4) = A - πR^2/2

Simplifying the expression:
- We can multiply the numerator and denominator of the second term by 2 to get a common denominator.
- Remaining area = (2A - πR^2)/2

Final answer:
- The remaining area of the sheet is (2A - πR^2)/2 = A - πR^2/2 = A/2

Therefore, the correct answer is option A) A/2.