PSAT Geometry — area and volume with the reference sheet

Area and volume is the first skill College Board lists in the PSAT/NMSQT's Geometry and Trigonometry domain, described as solving problems associated with length, area, volume and scale factors. The test gives you a reference sheet with the formulas for the rectangle, triangle, circle, rectangular prism, cylinder, sphere, cone and pyramid, so what it examines is choosing the right one, reading the figure correctly and not losing a factor along the way. That is what this material practices.

The twelve quiz questions each use a different formula or idea. Two are composite areas: a floor with a closet cut out of one corner, and a garden with a right-triangular patio in a corner, where forgetting the one-half of the triangle formula is the option waiting for you. Two run formulas backward, finding a triangle's height from its area and a cylinder's height from its volume. Volumes follow for a rectangular tank, a cone, a square pyramid and a spherical tank; the sphere question offers the sphere's surface area as a distractor, since the two formulas are easy to swap. One question states the cylinder's volume and asks for the cone with the same base and height, which is the one-third relationship behind both the cone and pyramid formulas. The surface area of a box separates square units from cubic ones. The last two turn on scale factors: an enlargement by 3 multiplies an area by 9, and a model at half size has one eighth of the volume.

There are no arcs or sectors here, because those belong to the circles material that College Board lists for the SAT. Circles appear only as the base of a cylinder or cone, and in the sphere.

The flashcards list every formula on the reference sheet except the circle's, plus the surface area of a box, composite figures, and the three scale-factor rules for length, area and volume.

The printable written work has eight open problems: an L-shaped patio with a paving cost in dollars, a trapezoid split into a rectangle and a triangle, a funnel and a jar of equal volume, three balls packed in a can with the empty space computed, two similar containers compared by label area and by capacity, a pyramid's volume and slant height, a shipping box's cardboard and capacity, and a pool's depth before and after a 20 percent change.

The oral exam asks one question at a time and wants you to name the formula before you use it.

Every figure is invented for practice, and nothing is taken from any published test.

• Choose the correct reference-sheet formula for a rectangle, triangle, prism, cylinder, cone, pyramid or sphere
• Find the area of composite figures by adding pieces or subtracting a cut-out
• Solve an area or volume formula for a missing dimension
• Use the one-third relationship between a cone and a cylinder, or a pyramid and a prism
• Tell a sphere's volume from its surface area
• Compute the surface area of a rectangular box
• Apply scale factors: lengths by k, areas by k squared, volumes by k cubed

Practice material written by Zestly, based on the PSAT/NMSQT Math skill "Area and volume" as described in College Board's Fall 2026 PSAT/NMSQT Student Guide and the Assessment Framework for the Digital SAT Suite (version 3.01, August 2024), whose Appendix D reproduces the reference sheet of formulas used here. All figures are invented. Zestly is not affiliated with College Board.