ACT Plane Geometry & Trigonometry: Formulas Worth Memorizing

Geometry and trigonometry make up a meaningful share of ACT Math, and unlike some other exams, the ACT does not provide a formula reference page. Every formula you need has to already be in your head. The good news is the list is short and well-defined, and once it's memorized, this topic becomes mostly about careful setup.

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Did you know

This is the one ACT math topic where memorization work pays off more directly than practice volume. A student who has the area, volume, and trig formulas cold will consistently outscore one who "sort of remembers" them and has to reconstruct them under time pressure.

What's tested

Four question types show up constantly. Each one below has the quick version, plus a "Learn more" with the specific formulas.

Area and perimeter

Rectangles, triangles, circles, and composite shapes built from combinations of them.

Learn more: the core formulas and the composite-shape trick

Rectangle: Area = length × width. Triangle: Area = ½ × base × height. Circle: Area = πr², circumference = 2πr.
Composite shapes (an L-shape, a rectangle with a semicircle cut out) usually split into pieces you already know how to handle. Break the shape into simple rectangles, triangles, and circle-pieces, solve each separately, then add or subtract.
Watch for diameter versus radius. A question that gives you the diameter and a student who plugs it straight into πr² without halving it first is one of the most common careless errors on this topic.

Volume and surface area

Rectangular prisms, cylinders, cones, and spheres.

Learn more: the three-dimensional formulas to have memorized

Rectangular prism: Volume = length × width × height. Cylinder: Volume = πr²h.
Cone: Volume = ⅓πr²h. Sphere: Volume = 4/3πr³. These two are the ones students most often forget under pressure, worth extra repetition.
Surface area questions usually ask you to find the total area of every face. Break the solid into its individual faces, calculate each one's area, and add them up rather than trying to recall a single combined formula.

Triangles and the Pythagorean theorem

Right triangles, special triangles, and the relationships between angles and sides.

Learn more: special right triangles save real time

Pythagorean theorem: a² + b² = c², where c is the hypotenuse, the side opposite the right angle. Applies only to right triangles.
3-4-5 and 5-12-13 are the two most common Pythagorean triples. Recognizing them instantly, instead of squaring and adding by hand, saves real time on the test.
45-45-90 triangles have sides in the ratio x : x : x√2. 30-60-90 triangles have sides in the ratio x : x√3 : 2x. Memorizing these two ratios turns several steps of trig into a single lookup.
The angles in any triangle always sum to 180°. A surprising number of questions are solvable with just that fact plus one given angle.

Right-triangle trigonometry

Sine, cosine, and tangent, and using them to find a missing side or angle in a right triangle.

Learn more: SOH-CAH-TOA and reading the triangle correctly

SOH-CAH-TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. Applies only to right triangles.
"Opposite" and "adjacent" are relative to the angle you're working from, not fixed sides of the triangle. Re-identify which side is which every time the reference angle changes.
The hypotenuse is always across from the right angle and is always the longest side, a fast sanity check on whether your setup is right.
Word problems about angles of elevation or depression are right-triangle trig in disguise, draw the triangle first, label the given angle and side, then apply SOH-CAH-TOA.

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Gotcha

Here's the classic trap, almost word for word how it shows up: a cone and a cylinder with the same radius and height, and the question asks for the ratio of their volumes. Students who don't have the cone formula memorized often guess the ratio is 1:2. It's actually 1:3, since a cone's volume is exactly ⅓ of a cylinder's with matching dimensions. Without the formula memorized, there's no way to reason your way to the right ratio, it has to already be in your head.

A quick sanity check before you lock in an answer

Check that your answer's units and rough size make sense for the shape described. An "area" answer with cubic units, or a volume smaller than one of the shape's own linear dimensions, means a formula got mixed up somewhere.

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Quick tip

Sketch the shape and label every given value before doing any calculation, even for questions that already come with a diagram. Adding your own labels makes it much easier to see which formula the question is actually asking for.

Reading about the formulas is step one.

Ember has original Plane Geometry and Trigonometry questions, each with the reasoning explained the same way, including exactly which formula it's testing.

Practice this topic now