面積 and 体積: area and volume
Connect area and volume to their units without assuming equality in one guarantees equality in the other.
A two-dimensional extent
面積, read めんせき, means area. 部屋の面積は二十平方メートルです。 says that the room has an area of twenty square meters.
面積を比べました。 means comparing areas. Do not change the comparison into height or weight.
A three-dimensional extent
体積, read たいせき, means volume. 箱の体積を求めました。 means finding a box's volume.
一立方メートル means one cubic meter. Distinguish 平方 from 立方 to keep area units separate from volume units.
Equal areas and different volumes
底面の面積が同じでも、高さが違う直方体は体積が違います。 says that rectangular prisms with equal base areas but different heights have different volumes.
体積は分かりますが、中に何が入っているかは分かりません。 says that the volume is known but the contents are not. Size does not establish what is inside.
Are 平方メートル and 立方メートル the same unit?
Show the answer and why
No. The former is square meters for area; the latter is cubic meters for volume.