SAT Math — area, volume, and what scaling does to them

Two ideas do most of the work in this part of the test, and one of them is not a formula at all.

The first is that units say what kind of answer you are holding. Feet are a distance, square feet an area, cubic feet a volume, and an answer in the wrong one of the three is wrong before its arithmetic is examined. Several wrong options in this set are built to be caught that way and nowhere else: a perimeter offered as an area, a side length subtracted from an area, a base area added to a height. The explanations point at the units rather than at the arithmetic, because that is the check a learner can apply under time pressure without redoing the work.

The second is what happens when a shape is scaled, which is where intuition reliably fails. Triple every length of a square and its area does not triple, it grows ninefold, because an area is a product of two lengths and the factor applies to both. Double every length of a cube and its volume grows eightfold. In both questions the plain length factor sits among the options, because it is the first answer nearly everyone reaches for, and both explanations give a concrete pair of figures rather than restating the rule. The cube question also draws out the distinction worth carrying away: doubling a box makes it four times as expensive to paint and eight times as roomy, and those two numbers are different because a surface has two dimensions and a space has three.

The rest is the machinery, each piece supplied rather than assumed. A rectangle's perimeter is run backwards to recover a missing side. A triangle's area is half the parallelogram on the same base, which is why the halving is there and not a rule to be memorised. A cylinder's volume is given in the question as the base area times the height, so nothing has to be recalled and the reasoning stays visible — and the same reasoning covers any prism. Two figures have pieces cut out of them, where only the two areas matter and where the hole sits makes no difference.

Two more questions turn on rounding a whole-number answer, and they go in opposite directions: tiles divide the floor exactly, while the water left over at the bottom of a container still needs a bottle of its own however little of it there is.

Every room, banner, yard and container here is invented, and nothing is drawn from any official publication.

• Use the units of an answer to check whether it is a length, an area or a volume before checking the arithmetic
• Work a perimeter backwards to recover a missing side
• Find the area of a triangle and a parallelogram, and say why only one of them halves
• Find the volume of a box and of any prism from a base area and a height
• Predict what scaling every length does to an area and to a volume, and why the two factors differ
• Find a remaining area after a piece is cut out, and round a whole-number answer in the direction the situation requires

Written for this catalogue, with no source document. The content is the measurement material named in the College Board's own public description of the Geometry and Trigonometry domain — area and volume, including the effect of changing a figure's dimensions; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test presents much of this material with a figure on screen and supplies a formula sheet; here every figure is described in words and every formula beyond the area of a rectangle is stated in the question itself. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.