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Geometry
Shape, measurement, proof, coordinate geometry, transformations, and spatial reasoning.
Start with Analytic Geometry →38 complete · 38 total lessons
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01Analytic GeometryAnalytic geometry proves geometric facts by representing points, lines, curves, and transformations with algebra.Full lesson + sheet02Angle RelationshipsAngle relationships connect measures created by intersecting lines, parallel lines, and transversals. Vertical angles are congruent; adjacent angles forming a straight line are supplementary.Full lesson + sheet03AnglesAn angle measures rotation between two rays sharing a vertex.Full lesson + sheet04Arc LengthArc length is the distance along part of a circle’s circumference. It is the same fraction of $2\pi r$ as the central angle is of a full turn: $s=(\theta/360^\circ)2\pi r$ in degrees.Full lesson + sheet05AreaArea measures two-dimensional coverage and connects geometric formulas to decomposition, scale, design, construction, and optimization.Full lesson + sheet06Area of a CircleThe area enclosed by a circle of radius $r$ is $A=\pi r^2$. The square on $r$ reflects two-dimensional scaling: multiplying radius by $k$ multiplies area by $k^2$.Full lesson + sheet07ChordsA chord is a segment whose endpoints lie on a circle. A diameter is the longest chord because it passes through the centre.Full lesson + sheet08Circle EquationA circle with centre $(h,k)$ and radius $r>0$ has equation $(x-h)^2+(y-k)^2=r^2$. Every solution point is distance $r$ from the centre, so the equation is the distance formula with the square root removed.Full lesson + sheet09Circle TerminologyA circle is the set of points in a plane at a fixed distance from a centre; the filled region is a disk.Full lesson + sheet10CirclesA circle is the locus of all points in a plane at fixed distance $r>0$ from a centre. Its symmetry makes circumference and area depend only on radius.Full lesson + sheet11CircumferenceCircumference is the distance around a circle. It equals $C=2\pi r$ or $C=\pi d$, where $d=2r$.Full lesson + sheet12Conic SectionsConic sections are curves obtained by intersecting a double cone with a plane: circles, ellipses, parabolas, and hyperbolas.Full lesson + sheet13Coordinate GeometryCoordinate geometry uses algebraic formulas for slope, distance, midpoint, and lines to analyze shapes on the plane.Full lesson + sheet14DilationsA dilation scales every point from a fixed centre by a factor $k$. Distances from the centre are multiplied by $|k|$; angle measures and shape are preserved, so the image is similar to the original.Full lesson + sheet15Distance FormulaThe distance formula uses horizontal and vertical change with the Pythagorean theorem to measure straight-line separation between coordinates.Full lesson + sheet16EllipseAn ellipse is the set of points whose sum of distances to two fixed foci is constant. In standard axis-aligned form $(x-h)^2/a^2+(y-k)^2/b^2=1$, the larger denominator identifies the major semi-axis.Full lesson + sheet17HyperbolaA hyperbola is the set of points for which the absolute difference of distances to two foci is constant.Full lesson + sheet18Inscribed AnglesAn inscribed angle has its vertex on a circle and sides along chords. Its measure is half the measure of its intercepted arc.Full lesson + sheet19Measurement and GeometryMeasurement and geometry connect shapes, units, formulas, scale, and spatial reasoning to solve practical problems.Full lesson + sheet20Midpoint FormulaThe midpoint formula averages corresponding coordinates to locate the point exactly halfway between two endpoints.Full lesson + sheet21Parabola EquationA parabola is the set of points equidistant from a focus and a directrix. Standard form $(x-h)^2=4p(y-k)$ opens vertically, while $(y-k)^2=4p(x-h)$ opens horizontally.Full lesson + sheet22PerimeterPerimeter measures total boundary length and supports fencing, framing, distance, scale drawings, and geometric problem solving.Full lesson + sheet23Points Lines and PlanesPoint, line, and plane are foundational undefined terms described by their relationships.Full lesson + sheet24PolygonsA polygon is a closed plane figure made of finitely many straight segments meeting endpoint to endpoint, with standard simple polygons having noncrossing sides.Full lesson + sheet25Pythagorean TheoremIn a right triangle, the area of the square on the hypotenuse equals the combined areas of the squares on the legs.Full lesson + sheet26QuadrilateralsA quadrilateral is a simple polygon with four sides and interior-angle sum $360^\circ$.Full lesson + sheet27ReflectionsA reflection maps every point across a mirror line so that the line is the perpendicular bisector of the segment joining point and image.Full lesson + sheet28RotationsA rotation turns every point through the same directed angle about a fixed centre. Distance from the centre and all shape measurements are preserved, so rotations are rigid transformations and preserve orientation.Full lesson + sheet29Sector AreaA sector is the region bounded by two radii and their intercepted arc.Full lesson + sheet30Surface AreaSurface area measures the total exposed area of a three-dimensional object by accounting for every face and curved surface exactly once.Full lesson + sheet31TangentsA tangent line to a circle meets it at exactly one point of tangency and is perpendicular to the radius drawn to that point.Full lesson + sheet32TransformationsA geometric transformation maps each point of a figure to an image point.Full lesson + sheet33TranslationsA translation slides every point by the same vector. It preserves distances, angles, parallelism, orientation, and shape, so it is a rigid motion.Full lesson + sheet34Triangle ClassificationTriangles are classified independently by side lengths and angle measures. By sides: scalene has no equal sides, isosceles at least two, equilateral three.Full lesson + sheet35Triangle CongruenceCongruent triangles have all corresponding sides and angles equal; one can be mapped onto the other by rigid motions.Full lesson + sheet36Triangle SimilaritySimilar triangles have equal corresponding angles and proportional corresponding side lengths. They share shape but may differ in scale.Full lesson + sheet37TrianglesA triangle is a nondegenerate polygon with three sides and interior-angle sum $180^\circ$.Full lesson + sheet38VolumeVolume measures three-dimensional capacity in cubic units and connects base area, height, scale, and real containers.Full lesson + sheet
