How do I find the volume of a rectangular prism (cuboid)?
Volume is length × width × height. Keep every measurement in the same unit before multiplying — mixing centimetres and metres is the most common source of an answer that is out by a factor of a thousand or a million. Volume units are cubic, so converting afterwards means cubing the conversion factor: 1 m³ is 1,000,000 cm³, not 100.
How do I find the surface area of a rectangular prism (cuboid)?
Surface area is 2(lw + lh + wh). Work out each face separately and add them, rather than trying to apply a single remembered formula — it is slower but far more reliable, and it makes it obvious when a face should be excluded because it is not exposed.
What is the space diagonal, and why does it matter?
The longest straight line inside the box, from one corner to the opposite one: √(l² + w² + h²). It is the practical limit on what fits inside — a length of pipe or a rolled poster can exceed every individual side and still fit if it is shorter than the space diagonal. This is exactly the calculation removal firms and airlines apply.
How does surface area scale compared with volume?
Volume grows with the cube of size while surface area grows with the square, so doubling every dimension gives eight times the volume but only four times the surface. This square-cube law explains why large animals need proportionally thicker legs, why big buildings lose heat more slowly per unit volume, and why crushed ice melts faster than a block.
Is a cube just a special rectangular prism?
Yes — one where all three dimensions are equal. Among all rectangular prisms of a given volume, the cube has the smallest surface area, which is why it is the most material-efficient box shape. It is also why packaging tends toward cubic proportions when material cost dominates.
Why does my answer differ from the textbook's?
Usually one of three things. Rounding too early — carry full precision through the working and round only the final answer. Mixed units, which is the largest source of error. Or using a rounded value of π: use your calculator's π rather than 3.14, which introduces an error of about 0.05% that compounds when cubed.