Area and volume

This skill point covers everything the exam asks about the size of a figure: the area of a plane shape, the volume of a solid, the surface area of a solid, and the ratios that connect two figures of the same shape at different sizes. College Board publishes the domain weight — Geometry & Trigonometry is about 15% of the Math section, which its own specifications put at 5 to 7 of the 44 questions.

What College Board does not publish is how those 5 to 7 questions split across the domain's four skill points, or how they distribute between the first module and the harder second one. The widely repeated prep-industry claim that geometry clusters in the easier module is reverse-engineered from released tests — plausible, useful for planning, and not an official figure. Do not build a strategy that depends on it.

Every Math module opens with a reference sheet. It is the only place on the entire exam where formulas are handed to you, and the single most valuable thing to know about it is where it stops. Every volume formula the exam needs is printed. Not one surface-area formula is. Neither is the trapezoid, the parallelogram, the equilateral triangle, the arc of a circle, or the rule that governs what happens when a shape is scaled up — which is the fact the hardest items in this domain are built on.

Weak path: an item describes a grain silo — a cylinder with a dome on top — and you scan the sheet for a silo, find nothing, and start improvising. Strong path: you already know the sheet contains no composite shapes and never will, so you cut the solid into printed pieces, price each piece separately, and add. Same knowledge, different first move, about ninety seconds apart.

Weak path on a scaling item: compute the original volume, compute the new dimensions, compute the new volume, divide. Strong path: notice that the question only ever asked for a ratio, and that a ratio of volumes is the cube of a ratio of lengths — so no dimension needs to be known, and neither does the formula. Four lines against one, and the one-line version cannot make an arithmetic slip.

If you can say what 2πrh is the area of, and why a cone carries a one-third while a cylinder does not, skip ahead. If either question is unfamiliar, this is the highest-value block on the page — every item later assumes these objects, and the exam will not teach them to you.

AREA counts unit squares. A rectangle 5 units by 3 units holds 3 rows of 5 squares, which is why A = ℓw: you are multiplying rows by columns. Area therefore always carries SQUARE units — cm², in², ft² — and that exponent is a running check on whether you computed the right kind of thing.

TRIANGLE: A = ½bh, printed on the sheet. A triangle is exactly half of a rectangle built on the same base with the same height, which is where the ½ comes from. The h is the PERPENDICULAR distance from the base to the opposite vertex — the straight-down height — not the length of a slanted side. Diagrams routinely label both, and only one of them is the height.

PARALLELOGRAM: A = bh, not printed. Cut the triangle off one end and slide it to the other and you have a rectangle of the same base and height, so a parallelogram has the same area as that rectangle. TRAPEZOID: A = ½(b₁ + b₂)h, not printed — average the two parallel sides, then multiply by the perpendicular distance between them. Both formulas are short, neither is on the sheet, and the trapezoid in particular shows up as the cross-section of a prism.

CIRCLE: A = πr² and C = 2πr, both printed, both written in terms of the RADIUS. Problems state diameters far more often than radii, and every printed formula wants r. Build the habit of writing r = d/2 as your first line, before touching a formula — a diameter used where a radius belongs multiplies an area by 4 and a volume by 8, and the exam puts both of those wrong numbers in the choices on purpose.

VOLUME counts unit cubes, and carries CUBIC units. For any prism or cylinder — any solid with two identical parallel ends and straight sides — the volume is the area of one end times how far it extends: V = (area of the base) × (height). That single sentence covers both printed formulas, because ℓwh is just (ℓw) × h and πr²h is just (πr²) × h. The general form V = Bh is not printed, and it is the one worth carrying, because it handles prisms whose base is a triangle, a trapezoid or a hexagon — cases the sheet cannot help you with at all.

POINTED SOLIDS take a third. A cone is V = (1/3)πr²h and a rectangular pyramid is V = (1/3)ℓwh, both printed. The pattern is worth stating so the sheet stops being a list: if the solid tapers to a single point, its volume is one third of the prism with the same base and the same height. If it has two identical parallel ends, it does not. And the h in both is the perpendicular height from apex to base, never the slanted edge running up the side.

SPHERE: V = (4/3)πr³, printed. A HEMISPHERE is half of that, and no hemisphere formula appears anywhere on the sheet — you halve it yourself. Domes, bowls and rounded silo tops are all hemispheres wearing a costume.

SURFACE AREA is the total area of the outside of a solid, and NOTHING about it is printed. You build it by adding up faces. A closed box is 2(ℓw + ℓh + wh) — three pairs of matching rectangles. A closed cylinder is 2πr² + 2πrh: two circular ends, plus the label, which unrolls into a rectangle whose width is the circumference 2πr and whose height is h. A sphere is 4πr², the one formula here that has to be memorised outright with no derivation to fall back on. Surface area always carries SQUARE units even though it wraps a solid, which is the fastest check that you computed the thing that was asked for.

PRINTED, in full. Areas: circle A = πr² with circumference C = 2πr; rectangle A = ℓw; triangle A = ½bh. Right triangles: c² = a² + b², plus the two special triangles — 30-60-90 with sides x, x√3, 2x, and 45-45-90 with sides s, s, s√2. Volumes: rectangular solid ℓwh; right circular cylinder πr²h; sphere (4/3)πr³; right circular cone (1/3)πr²h; rectangular pyramid (1/3)ℓwh. Then three facts in words: there are 360 degrees of arc in a circle, 2π radians of arc in a circle, and 180 degrees in the angles of a triangle.

NOT PRINTED, and this list is the real content of this lesson. Any surface area at all — box, cylinder, cone, sphere, none of them. Trapezoid area ½(b₁ + b₂)h. Parallelogram area bh and rhombus area ½d₁d₂. Equilateral triangle area (s²√3)/4 and regular hexagon area (3√3/2)s². The general prism V = Bh. Arc length and sector area. The equation of a circle, (x − h)² + (y − k)² = r². The distance and midpoint formulas. The space diagonal of a box, √(ℓ² + w² + h²). Every trigonometric ratio. And the scaling rules in the next section — the most-tested unprinted fact in the domain.

How to actually use the sheet. Open it once at the start of the module, look at it, close it. It lives behind a button on a screen, and a student who opens it six times has spent a question's worth of time on lookups. The volume formulas are worth knowing cold regardless: the sheet is insurance against a blank, not a method for working.

The habit the sheet quietly demands: every printed circular formula is written in terms of a RADIUS, and problems overwhelmingly state a DIAMETER. Before you touch a formula, write down r explicitly. This one line prevents the highest-frequency error in the domain, and it costs two seconds.

One sentence carries most of the hard items in this domain. If every linear dimension of a figure is multiplied by k, then every length is multiplied by k, every area — including surface area — is multiplied by k², and every volume is multiplied by k³. It holds for any shape whatsoever, including shapes with no formula, because it is a fact about multiplication rather than a fact about geometry.

It runs backwards, and backwards is where the harder items live. Areas in the ratio 9 : 1 means lengths in the ratio 3 : 1. Volumes in the ratio 8 : 27 means lengths in the ratio 2 : 3. Coming back from an area, take the square root; coming back from a volume, take the cube root. A cube of volume 3,375 cubic centimetres has an edge of ∛3,375 = 15 cm, and no formula beyond that is needed.

The word that licences all of this is SIMILAR. Similar figures have every dimension scaled by the same k. When only one dimension changes, the rule does not apply and you go back to the formula: triple only the radius of a cylinder and the volume is multiplied by 9; triple only the height and it is multiplied by 3; triple both and you get 27. The exam writes all three of those numbers into the same answer set.

Percentages are the most-missed form of this. Increasing every edge of a cube by 20% multiplies the volume by 1.2³ = 1.728 — a 72.8% increase, not 20% and not 60%. Percentage changes compound across dimensions; they do not add. The same fact in reverse: shrinking every dimension by 10% leaves 0.9³ = 0.729 of the volume, a 27.1% loss, not a 30% one.

Unit conversion is scaling wearing a different hat, and it is the most reliable place in the domain to lose a point. 1 foot = 12 inches, so 1 square foot = 144 square inches and 1 cubic foot = 1,728 cubic inches. 1 yard = 3 feet, so 1 cubic yard = 27 cubic feet. 1 metre = 100 centimetres, so 1 cubic metre = 1,000,000 cubic centimetres. Convert the LENGTHS first and compute once, or compute first and convert with the cubed factor — but never apply a linear factor to a volume, and never trust a conversion whose exponent you did not consciously choose.

At this point every printed formula is automatic and k³ is second nature. Six things are still moving points: hollowed solids, cross-sections, scaling run backwards, inscribed figures, the off-sheet formulas nobody warns you about, and a short list of execution errors that survive knowing all of the above.

HOLLOWED SOLIDS AND SUBTRACTION. A composite is not always an addition. A pipe is a cylinder minus a cylinder; a washer is a disc minus a disc; a box with a hole drilled through it is a box minus a cylinder. The efficient form for anything with a uniform cross-section is V = (cross-sectional area) × (length), so a pipe of outer radius R and inner radius r and length L is π(R² − r²)L in one line. The error the item is built to catch is subtracting the radii before squaring them: (R − r)² is not R² − r², and for R = 4, r = 3 the two differ by a factor of seven. Square first, subtract second — always.

CROSS-SECTIONS AND NETS. Two presentations recur. A NET is a solid unfolded flat: its area IS the surface area of the solid, so a net question is an addition problem, not a spatial one — count the faces on the page and add them. A CROSS-SECTION is a slice: the cross-section of a cylinder parallel to its base is a circle of radius r, and a slice through the axis is a rectangle 2r by h. Any solid with a constant cross-section is (that area) × (length), which is how you handle a prism whose base is a trapezoid, a hexagon, or an L-shape the sheet has never heard of.

SCALING BACKWARDS, AND NON-INTEGER FACTORS. Given a volume ratio, take the cube root to recover the linear ratio; given an area ratio, take the square root. Volumes of 54 and 128 are in the ratio 27 : 64, so the lengths are in the ratio 3 : 4. The same machinery answers questions with no ratio in sight: a cube of volume 3,375 cm³ has edge ∛3,375 = 15 cm and surface area 6(15²) = 1,350 cm². And the percentage form once more, because it is where 1500-level students actually lose this: +20% per edge is ×1.728 on volume, −10% per edge is ×0.729, and "the volume increased by 33.1%" means each edge grew by 10%, since 1.1³ = 1.331.

INSCRIBED FIGURES AND THE SPACE DIAGONAL. A sphere inscribed in a cube touches all six faces, so the sphere's DIAMETER equals the cube's edge — and the sphere fills πs³/6 ÷ s³ ≈ 52% of the cube, a useful sanity bound. A cube inscribed in a sphere touches at its corners, so the cube's SPACE DIAGONAL equals the sphere's diameter, and the space diagonal of a box is √(ℓ² + w² + h²) — Pythagoras applied twice, once across the base and once up. For a cube that is s√3. None of this is on the sheet, and "inscribed" is the word that tells you which of the two relationships is in play.

THE OFF-SHEET FORMULAS WORTH CARRYING. Trapezoid ½(b₁ + b₂)h. Parallelogram bh. Rhombus or kite ½d₁d₂. Equilateral triangle (s²√3)/4, which falls straight out of a 30-60-90 split: the height of an equilateral triangle of side s is (s√3)/2, so the area is ½ · s · (s√3)/2. Regular hexagon (3√3/2)s², because a regular hexagon is six equilateral triangles meeting at the centre. Closed box surface area 2(ℓw + ℓh + wh). Closed cylinder 2πr² + 2πrh. Sphere surface area 4πr². Sector area (θ/360)πr² and arc length (θ/360)(2πr) in degrees, or ½r²θ and rθ in radians. Nine formulas, none printed, and between them they cover every figure the exam uses that the sheet omits.

DENSITY, RATE AND CAPACITY, which is how hard area-and-volume items usually disguise themselves. Mass = density × volume, so a solid's weight is a geometry problem with one multiplication attached. A tank filling at a constant rate is volume ÷ rate = time, and the quantity that matters is almost always the volume ADDED, not the total volume — a level rising from 12 inches to 20 inches is eight inches of new water, and an answer computed from twenty inches is arithmetically flawless and wrong. Read whether the question is about a state or about a change before computing anything.

EXECUTION ERRORS, which is where the last few points genuinely sit. Reporting r² because the equation ended there. Forgetting to halve the sphere for a hemisphere. Adding the dome's radius to the cylinder's height when the diagram never said to. Leaving π symbolic when the question asked for a decimal, or evaluating it when the question said "in terms of π". Mixing units inside a single composite — a length in feet multiplied by a radius in inches. Answering in square units what was asked in cubic. And a format rule that is College Board's own: π cannot be typed into a student-produced response field, so a grid-in on this skill is always designed so that π cancels or so that a decimal is wanted — and if you are entering a decimal, the field must be filled, since a truncated repeating decimal is scored as wrong.