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Welcome to NRICH.

Geometry

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For measures, see under number. For coordinate geometry, see under algebra.

Shapes and terminology

• 2 Triangular prisms
• 2 Polygon names
• 2 Regular 2-D shapes
• 2 Septagons - are they real?
• 2 Polygon with 7 sides
• 2 11 sided shape
• 2 Regular shapes: definition
• 2 Types of triangle
• 3 Tessellating quadrilaterals
• 3 How many sides has a circle?
• 3 Parallel Lines
• 3 Article: Squareo'scope determines the kind of triangle
• 2 Three pyramids making a cube
• 3 Pyramid constructions
• 3 Nets of a cube
• 3 Names of polyhedra
• 3 Article: Classifying Solids using Angle Deficiency
• 4 Platonic solids; why there are only 5 regular polyhedra
• 4 Article: Impossible Polyhedra
• 4 Tesselating tetrahedra...
• 4 Colouring an octahedron
• 4 Article: The Cyclic Quadrilateral
• 5 Similarity
• 4 3D Shapes and Summation
• 5 Polygon diagonals

Symmetry

• 2 Order of Rotation
• 3 Rotational Symmetry
• 3 Symmetries of a cube
• 3 Article: The Frieze Tree
• 3 Article: Frieze Patterns in Cast Iron

Volume, area, perimeter, missing lengths

• 2 Circumference of a circle
• 3 Perimeter of a semi-circle
• 3 Area of a rhombus
• 3 Area of an ellipse (oval)
• 3 Volume of a sphere
• 3 Open box net problem
• 3 Article: Volume of a pyramid and a cone
• 3 Article: Volume of a mouhefanggai (and sphere)
• 4 Circumference problem
• 4 Height of equilateral triangle
• 4 Diagonal of a rectangle
• 4 Area of a hexagon
• 4 Area of triangle
• 4 Regular Polygon Areas
• 4 Regular polygons have largest area for a fixed perimeter
• 4 Finding a formula for stacking boxes
• 4 Surface Area of Prism
• 4 Ratio and Volume
• 4 Isoperimetric Quotients
• 4 Volume of a smartie
• 4 Surface Area of a Cylinder
• 4 Volume of dodecahedron & icosahedron
• 4 Grazing areas and sine/cosine rule
• 4 Area of a lune (and a segment)
• 4 Area of triangle & circumcircle
• 4 Graduation lines on a measuring cylinder
• 5 Circumference of an ellipse
• 5 Focus and circumference of an ellipse
• 5 Some geometrical ratios, and whether n-dim space is pointless
• 5 Volumes and areas of inspheres and incircles
• 5 Proving that regular shapes have maximum area proportional to the circumference
• 5 Circle ad infinitum problem
• 5 Length of chord from area of segment
• 5 Length of Chord in Circle
• 5 Intersection of two cylinders
• 5 Area enclosed by x=2cost, y=sint
• 5 Polygon Area

Pi (p)

• See under Number and Irrational Numbers for more about p
• 3 Pi
• 3 Circumference of a circle
• 5 Relationship between geometrical and series definitions for pi
• 6 Discussion of Volume and Surface Area

Pythagoras and trigonometry

• See under Algebra for Pythagorean Triples
• See under Trigonometric, exponential and logarithmic functions for more advanced trigonometry
• 4 What use is Pythagoras' Theorem
• 4 Pythagoras' theorem
• 4 Pythagoras' theorem
• 4 Different lengths of lines on a pinboard
• 4 Distances between posts in a triangular lattice
• 4 Pythagoras
• 4 Pythagoras' proof of Pythagoras' Theorem?
• 4 Leaning tower trigonometry problem
• 4 Grazing areas and sine/cosine rule
• 4 Area and perimeter of a triangle
• 4 Area to perimeter ratio of triangles with given angles
• 5 Triangle Geometry
• 5 An inequality for triangles
• 5 Triangle Areas
• 5 Right-angled Triangles and Perfect Squares
• 5 Distance of Earth from Sun

Angles

• 2 Why 360° in a full turn?
• 3 Proof that angles in a triangle add up to 180°
• 3 Angles of a pentagon problem
• 3 Tessellating quadrilaterals
• 3 Geometry problems
• 4 Triangle in a circle / angle in a semi-circle
• 5 Sum of angles in regular polygon
• 4 Article: The Cyclic Quadrilateral

Euclidean geometry

• 4 Paralellogram inside rectangle
• 4 Article: The Cyclic Quadrilateral
• 4 Opposite angles in a cyclic quadrilateral
• 4 Area proof with (or without) vectors
• 4 Cats Cradle Theorem
• 4 The inscribed circle
• 4 Circle Inscribed in a Triangle
• 4 Arbelos' theorem
• 4 Wellington's Theorem
• 4 Article: Incircles
• 4 Inscribed and circumscribed circles
• 4 Chords of a Circle
• 4 Common tangent problem
• 4 Ceva's theorem
• 4 Circles problem - pedal line
• 4 Circle tangent to circular arcs
• 4 Circles/tangents/angles proof
• 4 Intersecting chords theorem
• 5 Circle Tangents
• 5 Tangent to a circle
• 5 Circle Tangent to Ellipse
• 5 Shortest Distance between Skew Lines
• 5 Formulae for Circumcentre
• 5 How big is the smallest circle that covers a polygon?
• 5 Equilateral triangle and inscribed circle
• 5 180 degrees in a triangle
• 5 Volumes and areas of inspheres and incircles
• 5 Nice regular polygon theorem
• 5 Napolean's Triangle
• 5 Area of triangle & circumcircle
• 4 Three touching circles between parallel lines
• 5 Annoying geometry problem
• 5 Parallel pentagon problem
• 5 Triangle right-angled iff a²+b²+c²=8R²
• 5 Relationship between geometrical and series definitions for pi
• 5 Ellipse Area
• 5 Reflecting in ellipse
• 5 Loeffler Quadrilateral Theorem
• 5 Geometry in an affine space

Construction and locus

• 3 Article: What is the Circle Scribe Disk Compass?
• 4 Locus
• 4 Tethered horses and barns
• 4 Old goat or buffalo problem
• 4 Finding the centre of a circle
• 4 How can one draw an ellipse
• 4 Article: A Rolling Disc - Periodic Motion
• 5 Dividing angles with curves
• 5 Logarithmic Spiral
• 5 Line and Circle Loci
• 5 Geometric Constructions

Vectors

• See under algebra for matrices.
• 4 What are vectors?
• 5 a.b=abcosq
• 5 Vector equations
• 5 Vector equations of planes/A>
• 5 Parallel/coincident vectors
• 5 Vector intersection
• 5 Perpendicular vectors
• 5 Finding vector perpendicular to two others
• 5 Vectors: foot of perpendicular from given point to given line
• 5 Area proof with (or without) vectors
• 5 Parallelepiped inequality
• 5 Diagonals of rhombus perpendicular
• 5 Intersection of line & planes
• 5 Intersection of lines & planes
• 5 (r - a) × b = 0
• 5 Differentiating cross product

Topology

• 4 Topology: trefoil and torus
• 3 Article: Links and Knots
• 3 Article: Where do we get our feet wet?
• 4 Article: Euler's Formula and Topology
• 4 Motivate project:Euler's formula and topology
• 6 Number of topolgies and equivalence relations on a finite set
• 6 Poincaré Conjecture

Non-Euclidean geometry

• 3 Motivate project: Strange geometries
• 4 Non-Euclidean geometry
• 3 Article: How many geometries are there?
• 4 Articles: Geometry and Gravity

2D, 3D and beyond

• 3 2 dimensions and 3 dimensions
• 5 Conic Sections
• 5 Some geometrical ratios, and whether n-dim space is pointless
• 5 Four dimensions & equations of hyperplanes
• 5 Coordinates of unit hypersphere
• 5 Volumes and areas of inspheres and incircles
• 5 Hyperspheres
• 6 Negative dimensions
• 6 Volumes of n-spheres

Maths and art

• 0 Geometry in art
• 3 Article: Links and Knots
• 3 Article: The Frieze Tree
• 3 Article: Frieze Patterns in Cast Iron

Miscellaneous

• 3 Article: Where do we get our feet wet?
• 3 Pentominos
• 3 Square Dissections
• 3 Puzzling Puzzles
• 4 Motivate project:Where in the world am I?
• 4 Article: A Knight's Journey
• 4 Article: Zooming in on the Squares
• 4 Lengths of random chords
• 4 Motivate project:Fractals and fractal dimension
• 5 Fractal Images
• 5 Iterated Function Systems (Fractals)
• 5 Fractals: symmetrical roots
• 5 Chords of a circle
• 5 Transformations and matrices
• 5 See under complex numbers for some geometry problems involving complex numbers
• 5 Covering worms geometry problem
• 5 How close are these 15 points?
• 5 Lattice Points inside Circles
• 5 Radius of Curvature
• 5 Curves that slide along themselves
• 5 Plane dissection
• 5 Infinite Square Grid Problem
• 5 Intersecting Polygons
• 5 Regions in a Polygon
• 5 Lattice Points inside Circles
• 5 Calculate distance given Latitude and longitude
• 5 Shortest distance out of a wood
• 5 Combinatorics of polygon diagonals
• 6 Equidistant points on sphere
• 6 Quadric surfaces
• 6 Area of the Mandelbrot Set
• 6 Descartes' 4 Circle Theorem
• 6 Cantor Set