Area and volume questions on the ACT are pure recall plus careful arithmetic, so the topic comes down to memorizing about ten formulas and then reading the question slowly enough to use the right one. The ACT does not hand you a formula sheet, which sounds harsh but is good news: the list is short and finite, and once it is memorized these become some of the most predictable points available.
The flat shapes worth memorizing
- Rectangle: length times width. A square is the special case where both are the same, so its area is a side squared.
- Triangle: one half the base times the height.
- Parallelogram: the base times the height. A base of 9 and a height of 4 gives 36.
- Trapezoid: one half the sum of the two parallel sides, times the height. With parallel sides of 8 and 14 and a height of 5, that is one half times 22 times 5, which is 55.
- Circle: pi times the radius squared, with a circumference of 2 pi times the radius. More on those in how to solve ACT circle problems.
Height always means perpendicular height
In every one of those formulas, the height is measured at a right angle to the base. For a triangle or parallelogram drawn leaning over, the slanted side is longer than the height and the figure often labels both. Use the slanted side and you get a number that is too big, and it is reliably one of the choices. The trapezoid version of this trap is grabbing the two non-parallel sides, so find the parallel pair first.
The three solids that actually show up
- Rectangular box: length times width times height. A 4 by 5 by 6 box has a volume of 120.
- Cylinder: pi times the radius squared times the height, which is the circular base area stacked up. With a radius of 3 and a height of 10, the volume is 9 pi times 10, or 90 pi.
- Sphere: four thirds pi times the radius cubed. With a radius of 3, that is four thirds times pi times 27, or 36 pi.
A cone occasionally appears too, at one third pi times the radius squared times the height, which is one third of the cylinder with the same base and height. A question about a less common solid will often supply the formula, so read the stem before panicking.
Surface area is a different question
Volume measures how much fits inside; surface area measures how much wrapping paper you would need. For a rectangular box, the surface area is 2 times the quantity length times width, plus length times height, plus width times height. For the 4 by 5 by 6 box above, that is 2 times the quantity 20 plus 24 plus 30, or 148. A cylinder adds 2 pi times the radius squared for the two ends to 2 pi times the radius times the height for the side that unrolls into a rectangle, and a sphere is 4 pi times the radius squared. Ask whether the question is about filling something or covering it.
Composite figures: add or subtract
Harder questions glue shapes together. Break the figure into pieces you recognize, compute each, then add or subtract. Picture a 10 by 6 rectangle with a semicircle on one of the 6-unit ends. The semicircle has a diameter of 6, so its radius is 3, and half of 9 pi is 4.5 pi. The total area is 60 plus 4.5 pi, or about 74.1. Shaded regions call for subtraction instead: find the whole, find the hole, take the difference.
Units are where careful students still lose points
Watch for a problem that gives dimensions in one unit and asks for an answer in another. Conversion factors square for area and cube for volume, which is the part people miss. There are 12 inches in a foot, but 144 square inches in a square foot and 1,728 cubic inches in a cubic foot. A square yard is 9 square feet, not 3, so a floor measuring 12 feet by 9 feet covers 108 square feet, or 12 square yards.
Check that every dimension shares a unit before you multiply. A box listed as 2 feet by 18 inches by 1 foot needs the 18 inches rewritten as 1.5 feet, giving a volume of 3 cubic feet. The other classic mix-up is answering with perimeter when the question wants area, so underline the last few words before you start.
A quick self-check
Four to run through. A trapezoid has parallel sides of 6 and 10 and a height of 7; find the area. A cylinder has a radius of 5 and a height of 4; find the volume. A cube has an edge of 3; find its volume and surface area. A sphere has a radius of 6; find its volume. The answers are 56, since 8 times 7 is 56; 100 pi; a volume of 27 with a surface area of 54, since a cube has six faces of area 9; and 288 pi.
Flashcards get you to recognizing these formulas; timed practice gets you to using them without hesitating. Mixed geometry sets on thirty-six are good for that, and the full list of formulas to memorize for ACT Math collects everything in one place. For how this fits with the rest, see how much geometry is on the ACT.
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