ACT Mathematics — angles, triangles, circles and solids

Plane geometry on the ACT is a small set of rules asked in a large number of ways. This material covers that set: the angles two crossed lines make, what a transversal does across a pair of parallel lines, the angle sum and the exterior angle of a triangle, the Pythagorean theorem and the two special right triangles, the area formulas, circles, volume, and what happens to an area when every length is scaled.

Twelve questions, two on each of those six areas. Every figure is described completely in words, naming the points, because this is a text material with no diagram: "In triangle XYZ, angle X measures 40 degrees and angle Y measures 30 degrees, and side YZ is extended beyond Z." A reader can draw that and get the same figure the question is about, which is the constraint the whole bank is built to respect. Configurations that cannot be stated in a sentence or two are simply not used here.

The wrong options are the heart of it. Almost all of them are real quantities about the very same figure, correct arithmetic answering a question that was not asked — which is exactly how the section actually catches people. Asked for a circle's circumference, the option reading 49 pi is that circle's area. Asked for a cylinder's volume, the option reading 30 pi is the area of the curved surface wrapped around it. Asked for an exterior angle, the option reading 110 degrees is the interior angle at the same vertex. Each explanation names which quantity the wrong option is, so the lesson is not "that number is wrong" but "that number is the answer to a different question, and here is which".

Every key is checked inside its own explanation against something the figure forces: the three angles of the triangle are added back to 180, the hypotenuse is confirmed to be longer than either leg and shorter than the two of them walked end to end, the trapezoid's area is confirmed to fall between the two rectangles you could build on its parallel sides. Those checks are worth as much as the formulas, because they catch an answer that has gone wrong without your noticing.

Fourteen flashcards carry the rules, including the two special right triangles with the warning that the hypotenuse of a 30-60-90 is twice the SHORT leg specifically, and the pair on scaling: multiply every length by k and area goes up by k twice over, volume by k three times over, which is why a half-scale model holds one eighth as much.

Zestly is an independent study tool. It is not affiliated with ACT, Inc., which owns the test, and it is not an exam centre.

• Use vertical, supplementary and corresponding angles, and tell which of them a question is actually asking for
• Apply the triangle angle sum and the exterior-angle rule, and check the result by adding the three interior angles back to 180
• Apply the Pythagorean theorem and recognize the 45-45-90 and 30-60-90 side ratios
• Use the area formulas for a triangle, a rectangle and a trapezoid, with the height measured perpendicular to the base
• Tell a circle's circumference from its area, and halve a diameter before using any formula that wants a radius
• Find the volume of a box and a cylinder, and distinguish volume from curved surface area
• Scale an area by the square of the length factor and a volume by its cube
• Check a geometric answer against what the figure makes impossible, rather than trusting the arithmetic

Written for the geometry strand of the ACT mathematics section, following the categories the test's own published description names: properties of plane figures, angles and lines, the Pythagorean theorem, perimeter, area and volume, and the effect of scale on similar figures. Because this is a text material with no diagrams, every configuration is stated in full inside its own question, naming the points, so a reader can draw the figure and get the one the question means; configurations that cannot be described in a sentence or two were deliberately left out. Nothing is taken from any ACT publication or released test.