GEOMETRY
Triangles, circles, polygons, coordinate geometry, and a little trig — about a quarter of every paper.
What Geometry means on the AMC10
Geometry on the AMC10 rewards drawing carefully and seeing similar triangles. The recurring tools:
- Triangles — congruence, similarity, the Pythagorean theorem, special right triangles
- Circles — central and inscribed angles, power of a point, chords and tangents
- Polygons — interior angles, areas via decomposition
- Coordinate geometry — distance, slope, midpoint, intersections
- Three-dimensional figures — volume and surface area of prisms, pyramids, cones, cylinders, spheres
- Trigonometry basics — sine and cosine of common angles, the Law of Cosines when needed
If you find yourself doing heavy algebra on a geometry problem, stop and look for a similar-triangles shortcut. There almost always is one.
SUBTOPICS
Techniques for computing areas and perimeters of composite, shaded, and lattice regions.
Circumference, area, arcs, central and inscribed angles, tangent properties, power of a point, and the circle equation.
Distance, midpoint, slopes, line equations, area by shoelace, reflections, and point-to-line distance — the toolkit for placing geometry on the $xy$-plane.
Special quadrilaterals and their properties, angle sums, regular polygons, diagonal counts, and area techniques including the Shoelace formula.
Congruence criteria (SSS/SAS/ASA/AAS/HL), similarity criteria (AA/SAS/SSS), ratio chasing, and the k² area rule for similar figures.
Volume and surface area of prisms, cylinders, cones, spheres, and pyramids; space diagonals; cross-sections; and the effect of scaling on three-dimensional figures.
Triangle inequality, special right triangles, the Pythagorean theorem and triples, area formulas, and medians/altitudes.
SOH-CAH-TOA in right triangles, exact trig values at 30/45/60, the area formula (1/2)ab sin C, and the Law of Sines and Law of Cosines for general triangles.