To solve this problem, we need to determine the perimeter of figure A, given that the two figures have equal areas of 4 square centimeters.

Step 1: Calculate the area of Figure A

• Figure A is a square with both sides measuring 4 cm.
• The area of a square is given by:

$\text{Area of Figure A} = \text{side} \times \text{side} = 4 \, \text{cm} \times 4 \, \text{cm} = 16 \, \text{cm}^2$

However, the problem states that the two figures have equal areas of 4 square centimeters. This suggests that the side lengths of Figure A are incorrect in the image, or that we need to reconsider them. Let's use the given equal area information.

Step 2: Check the area of Figure B

• Figure B is a rectangle with dimensions 2 cm by 2 cm.
• The area of a rectangle is:

$\text{Area of Figure B} = \text{length} \times \text{width} = 2 \, \text{cm} \times 2 \, \text{cm} = 4 \, \text{cm}^2$

This confirms the area of both figures should indeed be 4 square centimeters.

Step 3: Adjust the side length of Figure A

Since Figure A is a square, its area must also be 4 square centimeters. Thus, the side length $s$ must satisfy:

$s^2 = 4 \quad \Rightarrow \quad s = \sqrt{4} = 2 \, \text{cm}$

So, the side length of Figure A is 2 cm, not 4 cm as originally labeled.

Step 4: Calculate the perimeter of Figure A

The perimeter $P$ of a square is given by:

$P = 4 \times \text{side length} = 4 \times 2 \, \text{cm} = 8 \, \text{cm}$

Final Answer:

The perimeter of Figure A is 8 cm.

Would you like further details on this? Here are a few related questions to consider:

1. How is the formula for the perimeter of a square derived?
2. Why was it necessary to adjust the side length of Figure A based on the given area?
3. What is the difference between calculating perimeter and area?
4. Can you apply the same logic to other shapes, like circles or triangles?
5. How would the solution change if Figure A were a rectangle instead of a square?

Tip: Always double-check the dimensions when areas are provided, as they can guide corrections to the problem's labels or assumptions!