Sphere — AMC & AIME Problems | LIVE by Po-Shen Loh

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Sphere — AMC & AIME Problems

Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses Sphere — a geometry concept — each with a professionally curated solution, many with a video solution too, from LIVE by Po-Shen Loh. Browse all concepts or the full past-contest archive.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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These problems’ difficulty ratings come from real solve attempts on our platform. Tap a band or a contest to filter the list below, and tap a selected band again to include all difficulties. See the full difficulty guide.

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AMC 10

2003 AMC 10B Problem 17 • 2004 AMC 10A Problem 25 • 2007 AMC 10A Problem 21 • 2013 AMC 10A Problem 22 • 2014 AMC 10B Problem 23 • 2017 AMC 10A Problem 11 • 2019 AMC 10A Problem 21 • 2021 AMC 10A Fall Problem 3 • 2021 AMC 10A Fall Problem 22 • 2023 AMC 10B Problem 20

AMC 12

1954 AMC 12 Problem 27 • 1977 AMC 12 Problem 27 • 1983 AMC 12 Problem 27 • 1999 AMC 12 Problem 29 • 2003 AMC 12B Problem 13 • 2004 AMC 12A Problem 22 • 2004 AMC 12B Problem 19 • 2005 AMC 12A Problem 22 • 2005 AMC 12B Problem 16 • 2013 AMC 12A Problem 18 • 2014 AMC 12A Problem 17 • 2014 AMC 12B Problem 19 • 2017 AMC 12A Problem 8 • 2018 AMC 12B Problem 23 • 2019 AMC 12A Problem 18

AIME

1987 AIME Problem 2 • 1998 AIME Problem 10 • 2000 AIME II Problem 12 • 2003 AIME I Problem 5 • 2020 AIME I Problem 6 • 2020 AIME II Problem 7 • 2021 AIME II Problem 10 • 2022 AIME I Problem 10 • 2022 AIME II Problem 3 • 2026 AIME I Problem 3 • 2026 AIME I Problem 12 • 2026 AIME II Problem 12

More geometry concepts

3D geometry altitude angle bisector angle bisector theorem angle chasing angle sum annulus arc area area decomposition area ratio Brahmagupta’s Formula British Flag Theorem centroid Ceva’s Theorem chord circle circle area circumcircle, circumcenter, and circumradius circumference cone congruence (geometry)coordinate geometry cube geometry cyclic quadrilateral cylinder diagonal distance formula ellipse equiangular polygon equilateral triangle Euler’s Polyhedron Formula Heron’s Formula homothety hyperbola incircle, incenter, and inradius inscribed angle isosceles triangle kite lattice point law of cosines law of sines mass points median (geometry)midpoint net (3D geometry)paper folding parabola parallel lines parallelogram perimeter perpendicular bisector Pick’s Theorem polyhedron power of a point power scaling of length, area, and volume Ptolemy’s Theorem pyramid Pythagorean Theorem Pythagorean Triple radical axis rectangle rectangular prism reflection (geometry)regular polygon rhombus right triangle sector shoelace formula similarity special right triangle square (geometry)Stewart’s Theorem surface area tangent circles tangent line transformation trapezoid triangle area triangle inequality trigonometric identity trigonometry vector Viviani’s Theorem volume