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Geometry

Learn how geometry describes space, shape, measurement, transformation, and structure.

For: Learners at different starting points, from first-time students to learners exploring advanced branches.

After this page you can: Choose a geometry learning path and use definitions, diagrams, and reasoning to solve and explain geometric problems.

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Part of: Mathematics

About Geometry

Geometry studies points, lines, planes, shapes, and the properties and relationships that describe space. Its topics range from familiar questions about angles, area, and volume to transformations, curved spaces, and discrete structures. OpenStax introduces points, lines, and planes, and asks learners to describe them using proper notation.

A useful way to study geometry is to define objects clearly, represent them with diagrams or coordinates, and then justify conclusions with reasoning or proof. Start with the foundations, then follow a goal into algebra, calculus, or linear algebra, which connect to different ways of describing geometric objects and change.

Start here

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I’m returning to geometry

Revisit transformations and symmetry through patterns.

Start with: 10.5 Tessellations

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I want to explore a broader branch

See how geometry can extend into topology and curved spaces.

Start with: Geometry and Topology in the Plane

Learn the foundations

1.Build a foundation in shapes and measurementBegin with points, lines, planes, angles, and the measurement of length, area, and volume.

Begin with points, lines, planes, angles, and the measurement of length, area, and volume.

Point

An idealized location with no width, length, or breadth.

Plane

A two-dimensional surface with length and width but no thickness.

Try it

In a coordinate plane, name the point three units right and two units up from the origin.

The point is (3, 2): the first coordinate gives the horizontal move and the second gives the vertical move.

2.Explore transformations and symmetryRigid motions such as translations, rotations, and reflections can move shapes while preserving their size and shape.

Rigid motions such as translations, rotations, and reflections can move shapes while preserving their size and shape.

Translation

A movement that shifts a shape in a specified direction.

Tessellation

A repeating pattern of shapes that fills a plane without gaps or overlaps.

Try it

A square moves four units right and one unit up. What transformation describes this movement?

It is a translation: each point of the square moves the same distance in the same direction.

3.Look beyond flat Euclidean spaceGeometry can study curved settings such as hyperbolic geometry, while topology examines features such as loops in the plane.

Geometry can study curved settings such as hyperbolic geometry, while topology examines features such as loops in the plane.

Hyperbolic geometry

A geometry that studies shapes and relationships in hyperbolic space.

Topology

A branch of mathematics concerned here with features such as loops in the plane.

Try it

Name one geometry topic and one topology topic listed in the MIT course description.

Hyperbolic geometry is a geometry topic, and loops in the plane are a topology topic.

4.Explore discrete and computational geometryDiscrete and computational geometry use geometric ideas to study finite structures and problems, including combinatorial geometry.

Discrete and computational geometry use geometric ideas to study finite structures and problems, including combinatorial geometry.

  • Geometric Combinatorics

    The course provides problem sets alongside its introduction to discrete and computational geometry.

Discrete geometry

A geometry area that includes the study of discrete and computational geometric problems.

Combinatorial geometry

A geometric area focused on methods in combinatorial geometry.

Try it

The MIT course introduces discrete and computational geometry and emphasizes a particular area. Name that area.

It emphasizes combinatorial geometry.

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Books

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Every free Geometry course and book on RedLectern

Free Geometry online courses

  1. Algebraic GeometryMIT 18.725 · Browse materials
  2. Algebraic GeometryMIT 18.725 · Browse materials
  3. Algebraic GeometryMIT 18.726 · Browse materials
  4. Computational GeometryMIT 2.158J · Browse materials
  5. Differential GeometryMIT 18.950 · Browse materials
  6. Geometric CombinatoricsMIT 18.319 · Browse materials
  7. Geometric Disciplines and Architecture Skills: Reciprocal MethodologiesMIT 4.105 · Browse materials · Video
  8. Geometric Folding Algorithms: Linkages, Origami, PolyhedraMIT 6.849 · Browse materials · Video
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  1. Geometry and Quantum Field TheoryMIT 18.238 · Browse materials
  2. Geometry of ManifoldsMIT 18.965 · Browse materials
  3. Geometry of ManifoldsMIT 18.966 · Browse materials
  4. Introduction to Arithmetic GeometryMIT 18.782 · Browse materials
  5. Seminar in Algebra and Number Theory: Computational Commutative Algebra and Algebraic GeometryMIT 18.704 · Browse materials
  6. Seminar in GeometryMIT 18.994 · Browse materials
  7. Topics in Algebraic Geometry: Algebraic SurfacesMIT 18.727 · Browse materials
  8. Topics in Algebraic Geometry: Intersection Theory on Moduli SpacesMIT 18.727 · Browse materials
  9. Topics in Geometry: Dirac GeometryMIT 18.969 · Browse materials
  10. Topics in Geometry: Mirror SymmetryMIT 18.969 · Browse materials
  11. Applied Geometric AlgebraMIT RES.8-001 · Open on OCW
  12. Double Affine Hecke Algebras in Representation Theory, Combinatorics, Geometry, and Mathematical PhysicsMIT 18.735 · Open on OCW
  13. Geometry and Topology in the PlaneMIT 18.900 · Open on OCW
How these courses are chosen

Listed as Geometry courses:

  • those whose title contains “Geometry” or “geometric”, or
  • those tagged “Geometry” or “geometric” as one of at most 2 topic tags, or as the first of 3.

Free Geometry textbooks and books

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