Area and volume: compute area of rectangles, triangles and circles, volume of prisms, cylinders, spheres, cones and pyramids, handle composite figures, and use the reference sheet.

What this skill is asking

Area and volume questions ask you to measure two-dimensional regions and three-dimensional solids. The Digital SAT (Geometry and Trigonometry domain) provides the common formulas on the reference sheet, so the skill is selecting the right formula, substituting correctly, and handling composite figures and scaling. The arithmetic is best done in Desmos.

The standard formulas

The reference sheet supplies these, so focus on choosing and using them.

A worked composite area

Composite figures are split into known pieces.

Composite figures and removed pieces

Most "find the shaded area" questions are composite: a region built from, or with a piece removed from, shapes you know. The reliable method is to decompose the figure into rectangles, triangles, and circles (or sectors), compute each, and add the pieces that are included or subtract the pieces that are removed. A classic is a circle inscribed in a square (shaded area $=$ square minus circle) or a rectangle with a semicircular end (rectangle plus half a circle). Identify the simple shapes, get each area from the reference sheet, and combine. The same decompose-and-combine idea works for the surface area or volume of a composite solid.

Scaling area and volume

A frequently tested and frequently missed idea is how area and volume respond to scaling. If you multiply every linear dimension of a figure by $k$, the area scales by $k^2$ and the volume by $k^3$, because area is two-dimensional and volume three-dimensional. So doubling the radius of a sphere multiplies its volume by $2^3 = 8$, and tripling the side of a square multiplies its area by $3^2 = 9$. Questions may phrase this as "if the dimensions are doubled, by what factor does the volume increase?" Recognising the $k^2$ and $k^3$ rules answers these without recomputing from scratch.

Working backward from a known area or volume

Many SAT questions run the formulas in reverse: they give the area or volume and ask for a dimension. The move is to substitute the known value and solve for the unknown. If a circle has area $36\pi$, then $\pi r^2 = 36\pi$, so $r^2 = 36$ and $r = 6$; the circumference is then $2\pi(6) = 12\pi$. If a cube has volume $64$, then $s^3 = 64$, so $s = 4$. These reverse questions often chain into a second step (find the radius, then the circumference; find the side, then the surface area), so solve cleanly for the dimension first, then use it. Because each formula is a single equation, working backward is just isolating the variable, the same algebra used elsewhere on the test, applied to a geometry formula.

Surface area of solids

Alongside volume, the SAT sometimes asks for surface area, the total area of all the faces of a solid. For a rectangular box this is the sum of the six rectangular faces, $2(\ell w + \ell h + w h)$; for a cylinder it is the two circular ends plus the curved side, $2\pi r^2 + 2\pi r h$. Surface area is not always on the reference sheet, so reason it out by adding the area of each face: unfold the solid in your mind into flat pieces, find each piece with a reference-sheet area formula, and sum. Keeping the difference clear, volume fills the inside (cubic units) while surface area wraps the outside (square units), prevents the common mix-up between the two.