Topic 03
Geometry
Shapes have rules — and they're prettier when you can grab the corners. Resize, rotate, slice and watch the numbers update in real time.
Two halves of geometry
Plane geometry covers flat shapes — lines, polygons, circles. Solid geometry moves to 3D — cubes, spheres, polyhedra. Everything below fits one or the other.
Featured playgrounds
Three quick demos to pick up before you dive into the topics.
Triangle explorer
Drag any corner. The angles always add to 180°.
Sum of angles
180°
always 180
Angle explorer
Click anywhere to point the second arm
Angle
60°
Acute angle
- Acute: less than 90°
- Right: exactly 90°
- Obtuse: between 90° and 180°
- Reflex: more than 180°
Circle playground
Drag the radius slider — watch every measurement update.
Radius (r)
80
Diameter (d)
160
Circumference
502.7≈ 2πr
Area
20106≈ πr²
Foundations
Where geometry starts: a single point, then a line, then a plane, then a solid.
Plane shapes
Flat figures you can draw on paper — triangles, quadrilaterals, polygons, circles.
Shapes
Circle, triangle, square, polygon — names, sides, properties.
Triangles
The three angles always sum to 180°. Drag to convince yourself.
Triangle centers
Centroid, circumcenter, incenter, orthocenter — four points every triangle hides.
Heron's formula
Area from three sides — no height needed.
Quadrilaterals
Square, rectangle, rhombus, parallelogram, trapezoid — the family tree.
Polygons
Three sides, four, five, six… and the angle each interior corner gets.
Circles
Radius, diameter, π, circumference, area — all in one playground.
Angles & lines
Lines meeting, parallels crossing, corners of every flavour.
Angles
Drag the arms of an angle and watch its measure update.
Parallel lines
Two parallels and a transversal — eight angles, three pair rules.
Using a protractor
Line up the baseline, read the right scale — measure any angle.
Radians
The other way to measure angles — and the one calculus prefers.
Slope
Rise over run — how steep is that line?
Measure
Edge length, surface inside, hypotenuse, sin/cos/tan — putting numbers on shapes.
Transform & symmetry
Slide, flip, turn, scale — and the patterns that look the same after.
Transformations
Slide, flip, turn and resize — see shapes move under each rule.
Symmetry
Reflection and rotational symmetry — fold the page.
Tessellations
Tiles that fit together with no gaps. Honeycombs, M.C. Escher, paving stones.
Congruence & similarity
Same shape, same size — vs. same shape, scaled.
Advanced plane geometry
Compass-and-straightedge tradition, the conic family, and circle theorems.
Constructions
Ruler and compass — the classical drafting moves.
Conic sections
Slice a cone and out fall ellipses, parabolas and hyperbolas.
Parabola
The curve a thrown ball traces — and what every satellite dish is shaped like.
Ellipse
A stretched circle. Planets, whispering galleries, racetracks.
Hyperbola
Two mirror-image curves. Comets, sonic booms, GPS triangulation.
Circle theorems
Inscribed, central, tangent — the family of circle laws.
Solid geometry
Three-dimensional space — polyhedra, prisms, pyramids, spheres, volume.
Solid geometry
Polyhedra, prisms, pyramids, Platonic solids — Euler's V−E+F = 2.
Cube
Six squares, twelve edges, eight corners — the simplest 3D solid.
Sphere
Every point the same distance from the center — the most efficient surface.
Cylinder
Two circles joined by a tube. Cans, pipes, drums.
Cone
A circle pulled to a point. Ice-cream, party hats, traffic markers.
Torus
A donut. One hole, but a surprising amount of geometry.
Prisms
Same shape, top and bottom — connected by rectangles.
Pyramids
A polygon base, all sides meeting at one peak.
Platonic solids
There are exactly five — tetrahedron, cube, octahedron, dodecahedron, icosahedron.
Euler's formula
V − E + F = 2. The hidden law every convex polyhedron obeys.
Cross-sections
Slice a 3D shape with a plane — what 2D shape falls out?
Nets
Unfold a solid into a flat net — a paper template you can cut and rebuild.
3D shapes & volume
Cubes, spheres, cylinders — drag a dimension and watch volume grow.
Real-world geometry
Where geometry shows up in the world around you.
Make something
Hands-on tools — drag, paint, click and build patterns of your own.
Tangram puzzle
Seven pieces, infinite shapes — drag and rotate them to fill the outline.
Symmetry artist
Paint with a mirrored brush and watch a kaleidoscope appear.
Spiral artist
Twist parameters and out come spirograph patterns, snail shells and galaxies.
Tessellation artist
Pick a tile, click to fill — make a pattern with no gaps.
Shape sorter
A drag-and-drop game for the youngest geometers.
Suggested path