Area and Volume
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalThe formulas here are given or easy; the setup is the test. The points leak in the slots, the triangle's half or the cone's third gone missing, the radius and height swapped, two dimensions multiplied where a volume needed three, in the power, tripled sides producing an area "three times bigger" or a squared ratio undone by halving, and at the end, in the measure: the perimeter answering an area question, the surface area answering a volume question.
These patterns aren't really about whether you can multiply dimensions. They're about whether every formula kept its coefficient and its squared or cubed slot, whether the scale factor got applied once per dimension, and whether the number you reported is the measure the question named.
The work
4 ways in · any order
Lesson
Area and Volume
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Write each formula with its coefficient and its squared or cubed slot before substituting, raise scale factors once per dimension and undo them with roots, and keep perimeter, area, and volume answers apart. The lesson works the method and the three patterns that derail it, and it saves the trap for last: the right figure's wrong measure.
Diagnostic
10-item topic check
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Ten items across the three patterns: formula slots and coefficients dropped, scale factors applied to the wrong power, and perimeters, areas, and volumes swapped at the finish. A mix of solids, similar figures, and perimeter-area conversions. Take it cold to surface the ones still catching you, or after the lesson to confirm they are gone.
Grid-in Check
Student-produced response
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About a quarter of SAT math answers are typed, not chosen, with no options to react to. These grid-in items diagnose by the value you enter, then route into the same drills the multiple-choice check feeds.
Targeted Practice
Drill a single pattern
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the pattern and you move on.