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Geometry

Triangles, circles, polygons, solids and coordinates.

经验教训

Core Triangles Heron's formula, the law of cosines and classifying triangles. triangle 3 4 5 Introductory Pythagorean theorem a² + b² = c² in a right triangle, and where it comes from. hypotenuse of 3 and 4 Introductory Circles Circumference, area and the meaning of π. circle radius 2 Core Polygons Interior angles, apothem and area of regular polygons. hexagon side 2 Core Solids Volume and surface area of spheres, cylinders, cones, boxes and pyramids. volume of a sphere with radius 3 Core Coordinate geometry Distance, midpoint, slope and the equation of a line. line through (0,1) and (2,5) Introductory Euclid's axioms and the structure of proof Points, lines, the five postulates, and why geometry was the first axiomatic science. triangle 5 5 5 Core Congruent and similar triangles SSS, SAS, ASA congruence, and similarity by scaling. triangle 3 4 5 Core Circle theorems Inscribed angles, tangents, chords, and the angle in a semicircle. circle radius 5 Core Transformations: translation, rotation, reflection, dilation Rigid motions and scalings, and the matrices that perform them. inverse of [[0,-1],[1,0]] Core Solid geometry: prisms, pyramids, spheres Volumes, surface areas and cross-sections in three dimensions. volume of a cone with radius 3 height 4

Shapes, measured. Give three sides and get every angle, the area and a drawn figure; give a radius and get a sphere you can spin. Coordinate geometry connects it all to algebra.

更多经验教训

Math Models and Geometry

Use a Problem Solving StrategySolve Money ApplicationsUse Properties of Angles, Triangles, and the Pythagorean TheoremUse Properties of Rectangles, Triangles, and TrapezoidsSolve Geometry Applications: Circles and Irregular FiguresSolve Geometry Applications: Volume and Surface AreaSolve a Formula for a Specific Variable

Geometry

Points, Lines, and PlanesAnglesPolygons, Perimeter, and CircumferenceTessellationsVolume and Surface AreaRight Triangle Trigonometry

Euclid

Euclid's ElementsEuclidean geometryAxiomStraightedge and compass constructionParallel postulateCongruence (geometry)Similarity (geometry)Thales's theoremAngle

Shapes

TriangleCirclePolygonQuadrilateralRegular polygonPolyhedronPlatonic solidSphereConeCylinderAreaVolumePerimeter

Coordinates and beyond

Analytic geometryCartesian coordinate systemDistanceConic sectionEllipseParabolaHyperbolaNon-Euclidean geometryHyperbolic geometrySpherical geometryProjective geometrySymmetryTessellationGolden ratio

其中一些经验教训从开放文本中改编: OpenStax Contemporary Mathematics, OpenStax Prealgebra 2e, Wikipedia. 每页课目都列出其来源和执照。

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人们对人们提出的问题

Why does every triangle have angles adding to 180°?

Draw a line through one vertex parallel to the opposite side: the two alternate angles equal the other two angles of the triangle, and the three angles at the vertex lie on a straight line. That is Euclid's proof, and it is why the fact holds only on a flat plane.

When do I use the law of sines versus the law of cosines?

Cosines when you know three sides, or two sides and the angle between them. Sines when you know an angle and the side opposite it, plus one more piece.

What is the difference between area and perimeter?

Perimeter is the length of the boundary (one dimension, measured in metres); area is the amount of surface inside (two dimensions, square metres). Doubling every side doubles the perimeter but quadruples the area.

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