In this image, we see two three-dimensional figures labeled as Figure A and Figure B. The questions ask to compare the volumes and surface areas of these two figures.

Part a: Volumes of the figures

The formula for the volume of a rectangular prism is $V = l \times w \times h$ (length × width × height).

• Figure A consists of 3 cubes stacked vertically, each cube with a side length of 1 cm. The volume of one cube is $1 \times 1 \times 1 = 1 \, \text{cm}^3$. Since there are 3 cubes in Figure A, the total volume is: $V_A = 3 \times 1 \, \text{cm}^3 = 3 \, \text{cm}^3$ However, the student mistakenly calculated $27 \, \text{cm}^3$, which is incorrect.

• Figure B consists of 1 cube. The volume of Figure B is: $V_B = 1 \times 1 \times 1 = 1 \, \text{cm}^3$ Again, the student calculated $9 \, \text{cm}^3$, which is incorrect as well.

Part b: Surface area comparison

Surface area is the total area of all the faces of a three-dimensional shape.

• Surface area of Figure A: It consists of three cubes arranged in an L-shape, and we need to count how many faces of each cube are exposed. Given the L-shape arrangement, it has more faces exposed compared to a simpler shape like Figure B.

• Surface area of Figure B: It consists of one single cube, so it has fewer exposed faces compared to Figure A. Thus, Figure A has a larger surface area.

Clarification

The student's reasoning that both the volume and surface area of Figure A are larger than those of Figure B is correct in concept, though there was a mistake in the volume calculation. The actual volumes should be:

• Volume of A: $3 \, \text{cm}^3$
• Volume of B: $1 \, \text{cm}^3$

Would you like me to explain these calculations in more detail or correct the surface area more precisely?

5 Related Questions

1. How is surface area calculated for composite shapes like Figure A?
2. What would be the volume if the cubes had different dimensions?
3. How would the comparison change if Figure B had more cubes?
4. What is the effect of adding more cubes on the surface area in each figure?
5. How can you determine the number of exposed faces in complex figures?

Tip:

When calculating the surface area of composite figures, be mindful of the hidden faces that do not contribute to the total surface area.