geometry
Sphere — AMC & AIME Problems
37 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.
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: Surface Area of a Sphere, Derivation (Proof) of Volume of a Sphere.
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.
AMC 10
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
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- Angle bisector theorem
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- Angle sum
- Annulus
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- Area decomposition
- Area ratio
- Brahmagupta’s Formula
- British Flag Theorem
- Centroid
- Ceva’s Theorem
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- Circle area
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- Coordinate geometry
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- Distance formula
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- Equilateral triangle
- Euler’s Polyhedron Formula
- Heron’s Formula
- Homothety
- Hyperbola
- Incircle, incenter, and inradius
- Inscribed angle
- Isosceles triangle
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- Lattice point
- Law of cosines
- Law of sines
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- 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)
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- Rhombus
- Right triangle
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- 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
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