Geometry
Core skills
Geometry on the ACT covers angle relationships, triangle properties (angle sum, isosceles, congruence, similarity), the Pythagorean theorem and special right triangles, perimeter and area of polygons and circles, arc length and sector area, volume and surface area of solids, coordinate geometry (distance, midpoint, slope, parallel and perpendicular lines), transformations, and right-triangle trigonometry including degree–radian conversion.
Traps to expect
- Inscribed versus central angles: an inscribed angle is half the central angle that subtends the same arc.
- Sector problems are proportions: area of a sector is $\frac{\theta}{360}\pi r^2$, and arc length is $\frac{\theta}{360}(2\pi r)$. The angle measure never replaces the circumference or area outright.
- Similar figures scale differently: lengths scale by $k$, areas by $k^2$, volumes by $k^3$.
- Perpendicular slopes are negative reciprocals, not merely negatives.
- Complementary (sum 90°) versus supplementary (sum 180°) are easy to swap under time pressure.
- Radians versus degrees: check the mode before computing trig values.
Worked example
A circle has radius 6 and a sector has central angle 60°. Its area is $\frac{60}{360}\pi(6^2) = \frac{1}{6}(36\pi) = 6\pi$. The corresponding arc length is $\frac{1}{6}(12\pi) = 2\pi$. Notice that the same fraction $\frac{1}{6}$ applies to both area and arc length.
Rule of thumb
On any figure, mark every length and angle you can derive before setting up an equation. The ACT rarely requires a clever insight - it requires using the two or three relationships already implied by the diagram.