△ Geometry

100 topics

• Points, Lines and Planes (Beginner) – The undefined building blocks of geometry.
• Line Segments and Rays (Beginner) – A segment has two endpoints; a ray starts at a point and goes on forever.
• Angles (Beginner) – Two rays sharing an endpoint; measured in degrees.
• Types of Angles (Beginner) – Acute <90°, right 90°, obtuse between 90° and 180°, straight 180°.
• Measuring Angles with a Protractor (Beginner) – Align the baseline and read the scale.
• Complementary and Supplementary (Beginner) – Pairs summing to 90° or 180°.
• Vertical Angles (Intermediate) – Opposite angles formed by intersecting lines are equal.
• Parallel Lines and Transversals (Intermediate) – Corresponding and alternate interior angles are equal.
• Triangles: Classification (Beginner) – By sides (equilateral, isosceles, scalene) and by angles.
• Triangle Angle Sum (Beginner) – Interior angles of a triangle add to 180°.
• Exterior Angle Theorem (Intermediate) – An exterior angle equals the sum of the two remote interior angles.
• Triangle Inequality (Intermediate) – Any two sides sum to more than the third.
• Congruent Triangles (Intermediate) – Same shape and size: SSS, SAS, ASA, AAS, HL.
• Similar Triangles (Intermediate) – Same shape, proportional sides: AA, SAS~, SSS~.
• Pythagorean Theorem (Intermediate) – In a right triangle the squares on the legs sum to the square on the hypotenuse.
• Pythagorean Triples (Intermediate) – Integer right triangles such as 3-4-5 and 5-12-13.
• Special Right Triangles (Intermediate) – 45-45-90 and 30-60-90 have fixed side ratios.
• Perimeter (Beginner) – Total distance around a shape.
• Area of Rectangle and Square (Beginner) – Length times width.
• Area of Parallelogram (Intermediate) – Base times perpendicular height.
• Area of Triangle (Intermediate) – Half base times height.
• Heron’s Formula (Advanced) – Triangle area from three sides.
• Area of Trapezoid (Intermediate) – Average of the parallel sides times height.
• Area of Rhombus and Kite (Intermediate) – Half the product of diagonals.
• Quadrilaterals (Intermediate) – Properties of squares, rectangles, rhombi, trapezoids and kites.
• Polygons (Intermediate) – Closed shapes with straight sides; regular polygons have equal sides and angles.
• Interior Angles of Polygons (Intermediate) – Sum of interior angles depends on number of sides.
• Exterior Angles of Polygons (Intermediate) – Exterior angles of any convex polygon sum to 360°.
• Regular Polygon Area (Advanced) – Use apothem and perimeter.
• Circles: Parts (Beginner) – Radius, diameter, chord, tangent, secant, arc and sector.
• Circumference (Intermediate) – Distance around a circle.
• Area of a Circle (Intermediate) – Area is π times radius squared.
• The Number Pi (Intermediate) – The ratio of circumference to diameter, ≈ 3.14159.
• Arc Length of a Circle (Advanced) – Part of the circumference.
• Sector Area (Advanced) – Part of the circle’s area.
• Central and Inscribed Angles (Advanced) – An inscribed angle is half the central angle on the same arc.
• Chord and Tangent Properties (Advanced) – A tangent is perpendicular to the radius at the point of contact.
• Power of a Point (Expert) – For a point and a circle, products of segment lengths along secants are equal.
• Cyclic Quadrilaterals (Expert) – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
• Volume of Prisms and Cylinders (Intermediate) – Base area times height.
• Surface Area of Prisms (Intermediate) – Sum the areas of all faces.
• Cylinders: Surface Area (Intermediate) – Two circles plus the curved face.
• Cones (Advanced) – Volume and surface area of a cone.
• Pyramids (Advanced) – Volume is one third base times height.
• Spheres (Advanced) – Volume and surface area of a sphere.
• Composite Solids (Advanced) – Break complex solids into simple ones.
• Platonic Solids (Advanced) – The five regular polyhedra.
• Euler’s Formula for Polyhedra (Advanced) – Vertices minus edges plus faces equals 2.
• Nets of Solids (Intermediate) – Unfolded flat patterns of 3D shapes.
• Cross Sections (Advanced) – Shapes formed when a plane slices a solid.
• Cavalieri’s Principle (Expert) – Solids with equal cross-sectional areas at every height have equal volume.
• Translations (Intermediate) – Slide a figure by a vector.
• Reflections (Intermediate) – Flip a figure across a line.
• Rotations (Intermediate) – Turn a figure around a point by an angle.
• Dilations (Intermediate) – Scale a figure from a centre by factor k.
• Symmetry (Intermediate) – Line and rotational symmetry of shapes.
• Tessellations (Advanced) – Tiling the plane with no gaps or overlaps.
• Congruence Transformations (Advanced) – Rigid motions preserve distances and angles.
• Distance Formula (Intermediate) – Distance between two points in the plane.
• Midpoint Formula (Intermediate) – The point halfway between two points.
• Equation of a Circle (Advanced) – All points at distance r from the centre.
• Coordinate Proofs (Advanced) – Prove geometric facts using coordinates.
• Section Formula (Advanced) – Point dividing a segment in ratio m:n.
• Area by Coordinates (Shoelace) (Expert) – Area of a polygon from vertices.
• Parabola (Geometric Definition) (Expert) – Points equidistant from a focus and a directrix.
• Ellipse (Expert) – Sum of distances to two foci is constant.
• Hyperbola (Expert) – Difference of distances to two foci is constant.
• Eccentricity of Conics (Expert) – Measures how stretched a conic is.
• Compass and Straightedge (Advanced) – Bisect angles, draw perpendiculars and regular polygons.
• Angle Bisector Theorem (Advanced) – A bisector divides the opposite side in proportion to adjacent sides.
• Medians and Centroid (Advanced) – Medians meet at the centroid, 2/3 along each.
• Circumcentre and Incentre (Advanced) – Centres of the circumscribed and inscribed circles.
• Orthocentre (Expert) – Intersection of triangle altitudes.
• Euler Line (Expert) – Centroid, circumcentre and orthocentre are collinear.
• Nine-Point Circle (Expert) – A circle through nine special points of a triangle.
• Ceva’s Theorem (Expert) – Condition for three cevians to be concurrent.
• Menelaus’ Theorem (Expert) – Condition for three points on sides to be collinear.
• Ptolemy’s Theorem (Expert) – In a cyclic quadrilateral the product of diagonals equals sum of products of opposite sides.
• Law of Sines in Geometry (Advanced) – Sides proportional to sines of opposite angles.
• Law of Cosines in Geometry (Advanced) – Generalised Pythagorean theorem.
• Vectors in Geometry (Advanced) – Represent displacements and prove geometry with vector algebra.
• Dot Product (Geometric) (Advanced) – Measures alignment of two vectors.
• Cross Product (3D) (Expert) – A vector perpendicular to two vectors.
• 3D Coordinates (Advanced) – Points (x,y,z) and distance in space.
• Lines and Planes in 3D (Expert) – Parametric and normal equations.
• Geometric Probability (Advanced) – Probability as ratio of areas or lengths.
• Golden Ratio (Advanced) – Ratio φ where the whole is to the larger part as the larger is to the smaller.
• Fractals (Expert) – Self-similar shapes with fractional dimension.
• Non-Euclidean Geometry (Expert) – Geometries where the parallel postulate fails: spherical and hyperbolic.
• Spherical Geometry (Expert) – Geometry on a sphere where lines are great circles.
• Projective Geometry (Expert) – Geometry with points at infinity; perspective and duality.
• Topology of Surfaces (Intro) (Expert) – Classify surfaces like torus and Möbius band by holes.
• Isoperimetric Problem (Expert) – Among all closed curves of given length, the circle encloses the most area.
• Locus Problems (Advanced) – Sets of points satisfying a geometric condition.
• Geometric Proofs (Intermediate) – Two-column and paragraph proofs using postulates and theorems.
• Euclid’s Postulates (Intermediate) – The five axioms of classical geometry.
• Mass Point Geometry (Expert) – A technique using weights to solve ratio problems in triangles.
• Inversion in a Circle (Expert) – A map sending P to P′ with OP·OP′ = r².
• Area and Volume Scaling (Advanced) – Lengths scale by k, areas by k², volumes by k³.
• Optimization in Geometry (Expert) – Maximise area or minimise distance using geometry and calculus.