Lessons
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.
More lessons
Math Models and Geometry
Geometry
Euclid
Shapes
Coordinates and beyond
Some of these lessons are adapted from open texts: OpenStax Contemporary Mathematics, OpenStax Prealgebra 2e, Wikipedia. Each lesson page names its source and licence.
Symbols used here
Tap any symbol for the full definition, a picture, and what every letter in it means.
Questions people ask
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.
Type it or draw it on the solver page: every step, a picture, the answer hidden until you ask. Solve →
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