ACT Mathematics — arcs, circle equations, similar figures and solids

ACT's Geometry subcategory covers shapes and solids, congruence and similarity, surface area and volume, and solving for missing values in triangles, circles and other figures, including equations of conic sections. ACT also states that recall of complex formulas is not required, so where a formula is not a basic one — the volume of a cone or a sphere — it is given in the question, as here.

The circle questions start with arc length and sector area as fractions of a full turn, then move to the coordinate plane: writing the equation of a circle from its center and radius, and recovering the center and radius from an expanded equation by completing the square twice. An inscribed-angle question checks that an angle with its vertex on the circle measures half its arc.

Similarity comes next: the classic shadow problem, a segment parallel to one side of a triangle where the whole side, not the remaining piece, sets the ratio, and two similar cylinders whose volumes scale with the cube of the length ratio. The solids questions compute the volume of a cone and of a ball from its diameter with the formulas supplied, and the surface areas of a closed box and a closed can, where the classic slips are counting each pair of faces once or forgetting one end.

The material offers two formats. The quiz has twelve multiple-choice questions with four answer choices each, as on the enhanced ACT, and every explanation names the slip behind each wrong choice, so a miss tells you what to fix. The flashcards collect the rules and formulas worth knowing by heart.

The content is based on ACT's published description of the mathematics section (reporting categories and the topics listed for each). It is independent practice and is not produced or endorsed by ACT.

• Compute arc lengths and sector areas from central angles
• Write the equation of a circle and find its center and radius by completing the square
• Apply the inscribed angle theorem
• Solve problems with similar triangles and similar solids
• Compute the volume and surface area of cones, spheres, boxes and cylinders

Practice material written by Zestly, based on ACT's published description of the ACT mathematics section (act.org, Mathematics Test Description: Preparing for Higher Math — Geometry). Formulas beyond the basics are given in the questions, as ACT states that recall of complex formulas is not required.