Geometry
Every lesson explains the concept, works through examples, and gives you practice problems with full solutions. Work through them in order, or jump to the topic you need.
46 lessons
- Sorting Shapes into CategoriesWhy shapes belong to more than one group at once, what makes a square also a rectangle, and how splitting a shape into equal parts gives each part a fraction name.
- Perimeter: The Distance Around a ShapeWhat perimeter means, how to find it for any shape by adding the sides, the rectangle shortcut, and how to work backwards from a known perimeter.
- Area of a Rectangle: Length Times WidthWhy the area of a rectangle is length times width, how to find a missing side, why area uses square units, and how to handle L-shaped figures.
- Splitting a Rectangle to Find Its AreaHow cutting a rectangle into two smaller rectangles keeps the total area the same, and how that turns a hard times fact into two facts you already know.
- Angles: Naming, Measuring and AddingWhat an angle is, why a degree is a fraction of a full turn, how to name angles as acute, right or obtuse, and why angle measures add when angles sit side by side.
- Parallel Lines, Perpendicular Lines and SymmetryHow to sort flat figures by parallel sides, perpendicular sides and angle size, why a right triangle is its own category, and how to find every line of symmetry.
- Classifying Quadrilaterals: The Shape HierarchyHow quadrilaterals are sorted by their properties, why every square is a rectangle, and how a shape inherits every attribute of the categories it belongs to.
- Volume of a Rectangular Prism: Length × Width × HeightWhy the volume of a box is length times width times height, why the units are cubed, and how the base-times-height form works for any prism.
- Area of a Triangle: Half the Base Times the HeightWhy the area of a triangle is half base times height, what counts as the height, and how to handle triangles where the height falls outside the shape.
- Volume of a Prism with Fractional Edge LengthsHow packing a box with unit-fraction cubes gives the same volume as multiplying the edge lengths, and how to apply V = lwh when the sides are fractions or mixed numbers.
- Surface Area and NetsWhat a net is, how unfolding a solid turns surface area into an addition of flat shapes, and how to find the surface area of a prism, cylinder and pyramid.
- Circumference of a Circle: C = πdHow to find the circumference of a circle from its diameter or radius, what pi actually is, and how to work backwards from a known circumference.
- Area of a Circle: A = πr²How to find the area of a circle from its radius or diameter, why the formula squares the radius, and the mistake of using the diameter instead.
- Scale Drawings and MapsHow to read a scale, convert between drawing and actual distances, find a scale factor, and why area scales by the square of the length factor.
- Angle Relationships: Complementary, Supplementary and Vertical AnglesWhat complementary, supplementary, vertical and adjacent angles are, why each relationship holds, and how to use them to find a missing angle.
- Angles in Triangles: The Angle Sum and Exterior AnglesWhy the angles of a triangle add to 180 degrees, what the exterior angle theorem says, and how parallel lines cut by a transversal produce equal angles.
- Transformations: Translations, Reflections and RotationsHow to translate, reflect and rotate a figure on the coordinate plane, the coordinate rule for each, and why all three leave lengths and angles unchanged.
- Congruence and Similarity: Dilations and Scale FactorWhat makes two figures congruent or similar, how a dilation resizes a figure about a center, what scale factor does to lengths and angles, and how to find a missing side.
- Drawing Triangles from Given Conditions, and Slicing SolidsWhich sets of side and angle measurements pin down exactly one triangle, which allow many, and which allow none — and what flat shapes appear when a solid is sliced.
- The Pythagorean Theorem: a² + b² = c²How to use a² + b² = c² to find a missing side of a right triangle, why c must be the hypotenuse, and how to find a leg rather than the hypotenuse.
- Distance on the Coordinate PlaneHow to find the distance between two points using the Pythagorean theorem, where the distance formula comes from, and how to measure a polygon drawn on a grid.
- Volume of Cylinders, Cones and SpheresThe volume formulas for a cylinder, cone and sphere, why a cone is exactly a third of its cylinder, and how to work backwards from a volume to a radius.
- Parallel and Perpendicular Lines: The Slope RulesParallel lines have equal slopes and perpendicular lines have negative reciprocal slopes — why both rules are true, and how to write equations using them.
- Rigid Motions and the Definition of CongruenceTransformations as functions on points, which ones preserve distance and angle, precise definitions of each rigid motion, and congruence defined by motion.
- Triangle Congruence: SSS, SAS, ASA and AASWhy three matching parts are enough to prove two triangles congruent, how each criterion follows from rigid motions, why SSA and AAA fail, and how CPCTC finishes a proof.
- Two-Column Proofs: Lines, Angles and Perpendicular BisectorsHow a geometry proof is built from definitions, postulates and earlier theorems, with complete proofs about vertical angles, parallel lines and perpendicular bisectors.
- Triangle Theorems: Angle Sum, Isosceles, Midsegment and MediansComplete proofs that a triangle's angles sum to 180°, that isosceles base angles are equal, that a midsegment is half the third side, and that the three medians meet.
- Parallelogram Proofs: Sides, Angles and DiagonalsProofs that a parallelogram's opposite sides and angles are congruent, that its diagonals bisect each other, and that equal diagonals make it a rectangle.
- Compass and Straightedge ConstructionsStep-by-step constructions for copying, bisecting, perpendiculars, parallels and inscribed polygons, with the congruence argument that proves each one works.
- Similar Triangles and the AA CriterionWhy two equal angles are enough to prove triangles similar, how corresponding sides stay in a fixed ratio, and how to use that ratio to measure things you cannot reach.
- Central and Inscribed Angles: The Inscribed Angle TheoremWhy an inscribed angle is half the central angle on the same arc, why an angle in a semicircle is a right angle, and why a tangent meets its radius at 90°.
- Arc Length, Sector Area and Radian MeasureWhy all circles are similar, why an arc's length is proportional to its radius, how that ratio defines the radian, and the formulas for arc length and sector area.
- Properties of Dilations: Parallel Lines and Scaled LengthsHow to dilate from any center, why a dilation sends a line to a parallel line and leaves lines through the center in place, and why every length scales by k.
- Partitioning a Segment in a Given RatioHow to find the point that divides a directed segment in a ratio such as 1:2, why the ratio becomes a fraction of the step from start to end, and why direction matters.
- The Equation of a CircleHow the Pythagorean theorem gives the equation of a circle, how to read the center and radius from it, and how completing the square recovers them from expanded form.
- Inscribed and Circumscribed CirclesConstructing the circles through and inside a triangle, why opposite angles of an inscribed quadrilateral add to 180°, and how to draw a tangent from an outside point.
- The Parabola from Its Focus and DirectrixWhy a parabola is every point equally far from a point and a line, how to derive its equation from that definition with the distance formula, and what the focus does.
- Trigonometric Ratios: Sine, Cosine and TangentWhy a side ratio in a right triangle depends only on the angle, what SOH CAH TOA names, why sine equals the cosine of the complement, and how to find a missing side.
- Area of a Triangle from Two Sides and an AngleWhy the area of any triangle is one half ab sin C, derived by dropping a height from a vertex, why the formula still works when the angle is obtuse, and how to use it.
- Solving Right Triangles: Inverse Trig and Angles of ElevationHow to find a missing angle with inverse sine, cosine and tangent, when the Pythagorean theorem is faster, and how to set up an elevation or depression problem.
- Special Right Triangles: 45-45-90 and 30-60-90Where the side ratios of the two special right triangles come from, how to use them without a calculator, and the exact sine, cosine and tangent values they give.
- The Law of Sines, and the Ambiguous CaseWhy each side of a triangle over the sine of its opposite angle gives the same number, how to solve triangles with it, and why SSA can give two triangles or none.
- The Law of CosinesWhy c² = a² + b² − 2ab cos C is the Pythagorean theorem with a correction, proved with coordinates, and how to find a side from SAS or an angle from SSS.
- Why the Volume Formulas WorkInformal arguments for circumference, the area of a circle, and the volumes of a cylinder, pyramid, cone and sphere, including Cavalieri's principle.
- Cross Sections and Solids of RevolutionWhat each plane cut through a cylinder, cone, sphere, cube and pyramid exposes, why every slice of a sphere is a circle, and which flat shapes sweep out which solids.
- Modeling with Geometry: Shapes, Density and DesignHow to replace a real object with a shape that answers the question, how density works in two and three dimensions, and how to solve a design problem under constraints.