Solid geometry and changes of scale

Imagine cutting the solid into layers

Place a rectangular box flat on the table. A layer one centimetre thick contains one cubic centimetre for every square centimetre of base. Stacking identical layers gives the volume (volume) as base area times height. Nothing requires the base to be a rectangle. The same argument works for a prism with a triangular end or a cylinder with a circular end. Pick two parallel ends of the same shape and measure the perpendicular separation between them.

A sloping edge can be much longer than that separation. Imagine pushing the top of a stack of cards sideways without changing the thickness of any card. The slanted stack occupies the same volume as the upright one because every horizontal layer still has the same area. If a diagram supplies a sloping length, look for a right triangle containing the perpendicular height before substituting into a formula.

A pyramid or cone tapers to a point. The formula for either is one third of the base area times the perpendicular height. The factor one third already accounts for all the shrinking cross-sections. In particular, halving the height remaining above a cut parallel to the base also halves each lateral dimension of the section there. Its area falls to one quarter. Treating the sections as if their areas decreased at a constant rate would give a different volume.

Decide which faces are on the boundary

Surface area (superficie) counts the material needed for a skin. Imagine painting every accessible face of the finished object. A rectangular box has three pairs of equal faces. Multiplying the two lengths along a face measures one member of a pair. Double the sum of those three different products for a closed box. If the box is open at the top, remove just the missing face.

A cylinder's curved wall unwraps into a rectangle whose long side runs around the rim. Its dimensions are the circumference and the height. Add two circular discs for a closed cylinder or just one for a pot without a lid. For a right circular cone the curved wall uses the slant height instead. The flattened wall is a sector with arc length equal to the base circumference, giving lateral area $\pi r\ell$ when the slant height is $\ell$. This length and the vertical height play different roles.

Scale one, two or three directions

Suppose a manufacturer makes a model with every length three times the corresponding original length. Along a straight edge three small rulers fit where one did before. Across a rectangular face there are three rows of three original rectangles. Inside a box there are three layers of nine original boxes. Accordingly a length factor $k$ gives an area factor $k^2$ and a volume factor $k^3$. Similar curved solids obey the same rules. For a sphere the formulas $4\pi r^2$ and $4\pi r^3/3$ display both powers directly.

To reverse the calculation, keep the direction of the ratio visible. A model with one eighth of the volume of a similar full-size object has half its lengths. If the question asks how much longer the original is, turn the ratio around after taking the cube root. Check whether the word “similar” is justified. Doubling just the height of a cylinder with fixed radius doubles its volume. Its curved wall doubles in area while its two ends keep their original areas.

Check the units before comparing

Changing the unit of measurement also applies a scale factor. An edge one metre long has length one hundred centimetres. A square of side one metre contains one hundred rows of one hundred square centimetres. A cubic metre contains one hundred such layers of thickness one centimetre. Convert all lengths to one unit before multiplying them, or convert the final answer with the appropriate square or cube factor. Write the physical unit beside the number even if the options omit it.