SAT Math — cones, spheres, pyramids and reverse scaling

The reference sheet in the digital SAT's testing app lists the volume of a cone, a sphere and a rectangular pyramid next to the familiar box and cylinder. The College Board's framework asks students to solve problems about the surface area or volume of a figure using given information such as length, area, surface area or volume, and to know that a scale factor of k multiplies lengths by k, areas by k² and volumes by k³. Having a formula on screen is not the same as knowing which number goes where, and the harder questions rarely hand over the height directly. This material practices both halves.

The twelve quiz questions start with the reference-sheet solids used in reverse or with a missing measurement: a cone whose slant height is given instead of its height, a sphere whose radius has to be recovered from its volume, a pyramid given by its base edge and lateral edge, where the height comes from half a diagonal, and a pyramid whose height follows from its volume. Surface area appears for a closed cylinder and for a cube known only by its volume. Two applied questions use volume as a quantity that moves: the rise in water level when a sphere is dropped into a cylindrical container, and the number of cone-shaped cups one gallon (231 cubic inches) can fill completely. A grain silo combines a cylinder with a hemisphere.

The last group works scale factors backward, which is where most errors happen. Students go from a volume ratio of 27 to a length factor of 3, from a surface-area ratio of 45 to 125 to a volume, and from a pyramid cut halfway up its height to the volume of the bottom piece, seven eighths of the whole. The distractors are the numbers students actually produce: the slant height used as the height, the factor one third dropped, a square root where a cube root belongs, the area factor used for a volume, a count rounded up to a cup that would not be full.

Five questions carry a drawing of the solid with the given measurements marked and the unknown left unlabeled: the cone, the pyramid with its lateral edge, the cylinder, the silo in side view and the container with the sphere inside. Every question also states its numbers in words.

The flashcards list the reference-sheet volume formulas, the surface area of a cylinder and a sphere, the slant-height relation in a cone, and the three scale-factor rules with worked ratio examples.

The material offers a quiz and flashcards. It is practice written by Zestly and is not affiliated with or endorsed by the College Board.

• Use the reference-sheet formulas for cones, spheres, pyramids and cylinders, including when the height must be found first
• Compute total surface area of cylinders and cubes
• Solve applied volume problems: displacement, filling containers, composite solids
• Work backward from a volume or surface-area ratio of similar solids to a length ratio
• Find the volume of the part of a pyramid below a parallel cut

Practice material written by Zestly, based on the College Board Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), Geometry and Trigonometry domain, skill Area and volume, and the Math reference sheet in the testing app.