geometry
3D geometry — AMC & AIME Problems
85 problems
Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses 3D geometry — 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.
LIVE focuses on challenge. As a social enterprise, we also explain standard school concepts one at a time, for free, on our sister site Expii. For this idea: Relating 2-D and 3-D Geometry, Concept of Volume, Length of Space Diagonal.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
Filter by difficulty
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.
AMC 8
AMC 10
- 2004 AMC 10A Problem 25
- 2006 AMC 10A Problem 24
- 2009 AMC 10B Problem 22
- 2010 AMC 10A Problem 17
- 2011 AMC 10A Problem 24
- 2012 AMC 10A Problem 21
- 2013 AMC 10A Problem 22
- 2014 AMC 10A Problem 19
- 2015 AMC 10A Problem 21
- 2020 AMC 10B Problem 20
- 2021 AMC 10A Spring Problem 13
- 2021 AMC 10A Fall Problem 22
- 2022 AMC 10A Problem 21
- 2023 AMC 10B Problem 20
AMC 12
- 1979 AMC 12 Problem 23
- 1980 AMC 12 Problem 26
- 1981 AMC 12 Problem 22
- 1982 AMC 12 Problem 16
- 1982 AMC 12 Problem 18
- 1999 AMC 12 Problem 29
- 2000 AMC 12 Problem 10
- 2001 AMC 12 Problem 15
- 2005 AMC 12A Problem 17
- 2005 AMC 12A Problem 25
- 2005 AMC 12B Problem 16
- 2007 AMC 12B Problem 25
- 2008 AMC 12A Problem 18
- 2009 AMC 12A Problem 22
- 2011 AMC 12A Problem 15
- 2011 AMC 12B Problem 18
- 2012 AMC 12A Problem 22
- 2013 AMC 12A Problem 18
- 2013 AMC 12B Problem 11
- 2014 AMC 12A Problem 17
- 2015 AMC 12A Problem 16
- 2016 AMC 12B Problem 13
- 2016 AMC 12B Problem 23
- 2018 AMC 12B Problem 23
- 2019 AMC 12B Problem 11
- 2019 AMC 12B Problem 18
- 2021 AMC 12B Spring Problem 14
- 2022 AMC 12A Problem 12
- 2024 AMC 12A Problem 24
- 2024 AMC 12B Problem 23
- 2025 AMC 12A Problem 20
AIME
- 1983 AIME Problem 11
- 1984 AIME Problem 9
- 1989 AIME Problem 12
- 1990 AIME Problem 14
- 1992 AIME Problem 7
- 1995 AIME Problem 12
- 1998 AIME Problem 10
- 1999 AIME Problem 15
- 2001 AIME I Problem 12
- 2002 AIME I Problem 15
- 2005 AIME II Problem 10
- 2006 AIME I Problem 14
- 2007 AIME I Problem 13
- 2008 AIME I Problem 15
- 2011 AIME I Problem 13
- 2013 AIME I Problem 7
- 2015 AIME I Problem 15
- 2015 AIME II Problem 9
- 2016 AIME I Problem 4
- 2016 AIME II Problem 14
- 2017 AIME II Problem 14
- 2017 AIME II Problem 15
- 2018 AIME I Problem 7
- 2021 AIME I Problem 6
- 2021 AIME II Problem 10
- 2022 AIME I Problem 10
- 2024 AIME I Problem 14
- 2024 AIME II Problem 8
- 2026 AIME I Problem 3
- 2026 AIME I Problem 12
- 2026 AIME II Problem 12
More geometry concepts
- 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
- Sphere
- 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