---
title: "Geometry"
description: "110 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/geometry
source: https://lightmysky.com/learn/mathematics/areas/geometry.md
retrieved: 2026-10-19
---

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# Geometry

110 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/geometry

- [2-D shapes](https://lightmysky.com/learn/mathematics/2-d-shapes-mt_KJeEeTutJI): Recognise and name common 2-D shapes (circles, triangles, rectangles including squares)
- [3-D shapes](https://lightmysky.com/learn/mathematics/3-d-shapes-mt_Qcp2d_kuta): Recognise and name common 3-D shapes (cubes, cuboids, pyramids, spheres, cylinders, cones)
- [Positional Language](https://lightmysky.com/learn/mathematics/positional-language-mt_qeZYF6HZ4o): Describe the position of objects using terms such as above, below, beside, in front of, behind, next to
- [Comparing 2-D and 3-D shapes](https://lightmysky.com/learn/mathematics/comparing-2-d-and-3-d-shapes-mt_yGv8doDAmp): Analyse and compare 2-D and 3-D shapes using informal language to describe sides, vertices, and other attributes
- [Building & Drawing Shapes](https://lightmysky.com/learn/mathematics/building-and-drawing-shapes-mt_2VR963szuk): Model shapes by building from components (e.g. sticks and clay balls) and by drawing
- [Combining Simple Shapes](https://lightmysky.com/learn/mathematics/combining-simple-shapes-mt_XjwUlmxdCT): Compose simple shapes to form larger shapes (e.g. two triangles make a rectangle)
- [Flat vs Solid Shapes](https://lightmysky.com/learn/mathematics/flat-vs-solid-shapes-mt_mnEVZNkX3p): Distinguish two-dimensional (flat) shapes from three-dimensional (solid) shapes
- [2-D faces on 3-D shapes](https://lightmysky.com/learn/mathematics/2-d-faces-on-3-d-shapes-mt_UooUHC_V7U): Identify 2-D shapes on the surface of 3-D shapes (e.g. a circle on a cylinder, a triangle on a pyramid)
- [Building with 3-D Shapes](https://lightmysky.com/learn/mathematics/building-with-3-d-shapes-mt_QNxFnxikCN): Compose three-dimensional shapes (cubes, right rectangular prisms, right circular cones, right circular cylinders) and create composite shapes; build new shapes from component shapes
- [Composing Shapes](https://lightmysky.com/learn/mathematics/composing-shapes-mt_MqqR7VUoz1): Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, quarter-circles) to create composite shapes, and compose new shapes from the composite shape
- [Defining attributes of shapes](https://lightmysky.com/learn/mathematics/defining-attributes-of-shapes-mt_bfhng6mOuy): Distinguish defining attributes of shapes (e.g. triangles are closed and three-sided) from non-defining attributes (e.g. colour, orientation, overall size)
- [2-D shapes (age 6+)](https://lightmysky.com/learn/mathematics/2-d-shapes-age-6-mt_sBcRdUfAzV): Identify and describe properties of 2-D shapes including the number of sides and line symmetry in a vertical line
- [Edges, vertices, and faces](https://lightmysky.com/learn/mathematics/edges-vertices-and-faces-mt_Mnodea7mG_): Identify and describe properties of 3-D shapes including the number of edges, vertices, and faces
- [Patterns & Sequences](https://lightmysky.com/learn/mathematics/patterns-and-sequences-mt_WXW0hjNhph): Order and arrange combinations of mathematical objects in patterns and sequences
- [Sorting 2-D and 3-D shapes](https://lightmysky.com/learn/mathematics/sorting-2-d-and-3-d-shapes-mt_fEuoBYw6bU): Compare and sort common 2-D and 3-D shapes and everyday objects by their properties
- [Drawing 2-D shapes and making 3-D shapes](https://lightmysky.com/learn/mathematics/drawing-2-d-shapes-and-making-3-d-shapes-mt_vJUa62bxeR): Draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them
- [Drawing shapes with given attributes](https://lightmysky.com/learn/mathematics/drawing-shapes-with-given-attributes-mt_zuOGOGFAKb): Recognise and draw shapes having specified attributes (e.g. a given number of angles or equal faces); identify triangles, quadrilaterals, pentagons, hexagons, and cubes
- [Classifying quadrilaterals](https://lightmysky.com/learn/mathematics/classifying-quadrilaterals-mt_Xt1cRqaBOW): Understand that shapes in different categories may share attributes defining a larger category; classify quadrilaterals (rhombuses, rectangles, squares) and draw examples of quadrilaterals not in those subcategories
- [Lines of symmetry in 2-D shapes](https://lightmysky.com/learn/mathematics/lines-of-symmetry-in-2-d-shapes-mt_hRZyKvz1KN): Identify lines of symmetry in 2-D shapes presented in different orientations; recognise line-symmetric figures and draw lines of symmetry
- [Completing symmetric figures](https://lightmysky.com/learn/mathematics/completing-symmetric-figures-mt_hR2Y7NhMSY): Complete a simple symmetric figure with respect to a specific line of symmetry
- [Nets of 3-D Shapes](https://lightmysky.com/learn/mathematics/nets-of-3-d-shapes-mt_vnJEztczji): Identify, draw, and interpret nets of common 3-D shapes — cubes, cuboids, triangular prisms, and square-based pyramids — by predicting which 3-D shape a given flat arrangement of faces will fold into, checking whether a net will close completely, and sketching a net from a description or 3-D model; understand the relationship between the number of faces and the structure of the net
- [3-D shapes (age 9+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-9-mt_wUSbRt3-qw): Identify 3-D shapes, including cubes and other cuboids, from 2-D representations
- [3-D shapes (age 10+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-10-mt_REwgr0d_ss): Recognise, describe, and build simple 3-D shapes, including making nets
- [3-D shapes (age 11+)](https://lightmysky.com/learn/mathematics/3-d-shapes-age-11-mt_nRksLqt-iR): Use the properties of faces, surfaces, edges, and vertices of 3-D shapes (cubes, cuboids, prisms, cylinders, pyramids, cones, and spheres) to solve problems, including visualising cross-sections
- [Constructions and Loci](https://lightmysky.com/learn/mathematics/constructions-and-loci-mt_yqoXwrKaW3): Construct perpendicular bisectors, angle bisectors and perpendiculars with compasses, and shade the region that fits a set of distance conditions.
- [Turns & Directions](https://lightmysky.com/learn/mathematics/turns-and-directions-mt_Y9XKzLrUAZ): Describe movement and direction, including whole, half, quarter, and three-quarter turns
- [Position, direction, and movement](https://lightmysky.com/learn/mathematics/position-direction-and-movement-mt_TdV9YGJEoY): Use mathematical vocabulary to describe position, direction, and movement, including straight lines and distinguishing rotation as a turn in terms of right angles (quarter, half, three-quarter turns, clockwise and anti-clockwise)
- [Understanding angles](https://lightmysky.com/learn/mathematics/understanding-angles-mt_8OAGVdeTJ_): Recognise angles as a property of shape or a description of a turn
- [Right Angles & Turns](https://lightmysky.com/learn/mathematics/right-angles-and-turns-mt_MFfYcnv6Tv): Identify right angles; recognise that two right angles make a half-turn, three make three-quarters, and four make a complete turn
- [Parallel and perpendicular lines](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-mt_u23IGDxOpk): Identify horizontal and vertical lines and pairs of perpendicular and parallel lines
- [First Quadrant Coordinates](https://lightmysky.com/learn/mathematics/first-quadrant-coordinates-mt_jBQS-CicNn): Describe positions on a 2-D grid as coordinates in the first quadrant
- [Coordinates (age 8+)](https://lightmysky.com/learn/mathematics/coordinates-age-8-mt_i5_HnoFOYw): Plot specified points on a coordinate grid and draw sides to complete a given polygon
- [Describing Movements](https://lightmysky.com/learn/mathematics/describing-movements-mt_Uq5vYqboCR): Describe movements between positions as translations of a given unit to the left/right and up/down
- [Types of angles](https://lightmysky.com/learn/mathematics/types-of-angles-mt_h0CVtqI2xo): Identify acute and obtuse angles; compare and order angles up to two right angles by size
- [Lines, Rays & Angles](https://lightmysky.com/learn/mathematics/lines-rays-and-angles-mt_QysgF57dxh): Draw and identify points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines; identify these in two-dimensional figures
- [Classifying shapes by line properties](https://lightmysky.com/learn/mathematics/classifying-shapes-by-line-properties-mt_r8c43QB6wx): Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or angles of a specified size; recognise right triangles as a category
- [Transformations on a grid](https://lightmysky.com/learn/mathematics/transformations-on-a-grid-mt_SH7QgFl8-v): Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size
- [Reflection and Translation on a Grid](https://lightmysky.com/learn/mathematics/reflection-and-translation-on-a-grid-mt_efOeaGFyGM): Identify, describe, and represent the position of a shape following a reflection or translation using appropriate language; know that the shape has not changed
- [What Is an Angle?](https://lightmysky.com/learn/mathematics/what-is-an-angle-mt_3S10OOGPqu): Understand that an angle is a geometric shape formed by two rays sharing a common endpoint (vertex); recognise angles in real-life contexts and 2-D shapes
- [Degrees and turns](https://lightmysky.com/learn/mathematics/degrees-and-turns-mt_qUGMyMYn9m): Know that angles are measured in degrees, where one degree is 1/360 of a full turn; understand that an angle turning through n one-degree angles has a measure of n degrees
- [Estimating Angles](https://lightmysky.com/learn/mathematics/estimating-angles-mt_YUJ5pwalqL): Estimate and compare acute, obtuse, and reflex angles in degrees; classify angles by type and order them by size
- [Geometric diagram conventions](https://lightmysky.com/learn/mathematics/geometric-diagram-conventions-mt_e4V6hvcuEJ): Use and interpret standard geometric diagram conventions: mark right angles with a small square, equal lengths with single or double tick marks, and equal angles with arc marks; label angles in three-letter notation (∠ABC) and individual angles with a single letter or number; draw diagrams showing angles at a point, angles on a straight line, and angles inside polygons with these conventions; read diagrams with these marks to identify given information and find unknown values
- [Angle Sum Rules](https://lightmysky.com/learn/mathematics/angle-sum-rules-mt_oqziWKry-L): Know that angles at a point sum to 360° (one whole turn), angles on a straight line sum to 180°, and vertically opposite angles are equal; use these facts to find missing angles
- [Measuring angles](https://lightmysky.com/learn/mathematics/measuring-angles-mt_4MFUAsbx_6): Measure angles in whole-number degrees using a protractor; draw given angles and sketch angles of specified measure
- [Regular and irregular polygons](https://lightmysky.com/learn/mathematics/regular-and-irregular-polygons-mt_8H2kO4k2B9): Distinguish between regular and irregular polygons based on reasoning about equal sides and equal angles
- [2-D shapes (age 10+)](https://lightmysky.com/learn/mathematics/2-d-shapes-age-10-mt_hdrMoiTgqu): Draw 2-D shapes using given dimensions and angles, using a ruler and protractor accurately
- [Classifying shapes by properties](https://lightmysky.com/learn/mathematics/classifying-shapes-by-properties-mt_liIW336odh): Compare and classify geometric shapes based on their properties and sizes; understand that attributes belonging to a category also belong to all subcategories; classify two-dimensional figures in a hierarchy based on properties
- [Angles in triangles (age 10+)](https://lightmysky.com/learn/mathematics/angles-in-triangles-age-10-mt_cdMlC7EpTJ): Find unknown angles in triangles, quadrilaterals, and regular polygons using angle sum properties
- [Parts of a circle](https://lightmysky.com/learn/mathematics/parts-of-a-circle-mt_xq3YHZ2zeR): Illustrate and name parts of circles, including radius, diameter, and circumference; know that the diameter is twice the radius
- [The coordinate plane and ordered pairs](https://lightmysky.com/learn/mathematics/the-coordinate-plane-and-ordered-pairs-mt_oDlduFnemk): Understand a coordinate system defined by two perpendicular number lines (axes) with an origin at (0,0); know that an ordered pair (x, y) specifies a unique point where the first number gives horizontal distance and the second gives vertical distance from the origin
- [Plotting points in the first quadrant](https://lightmysky.com/learn/mathematics/plotting-points-in-the-first-quadrant-mt_snlqRCiA1R): Plot and read ordered pairs in the first quadrant of the coordinate plane; represent real-world and mathematical problems by graphing points and interpreting coordinate values in context
- [Geometric terms and notation](https://lightmysky.com/learn/mathematics/geometric-terms-and-notation-mt_WtIFJSCQIT): Use conventional geometric terms and notation to describe, sketch, and draw points, lines, parallel and perpendicular lines, right angles, regular polygons, and reflectively/rotationally symmetric polygons
- [Properties of triangles, quadrilaterals and circles](https://lightmysky.com/learn/mathematics/properties-of-triangles-quadrilaterals-and-circles-mt_DNYQLahbfa): Derive and illustrate properties of triangles, quadrilaterals, and circles using appropriate language, including interior angles, diagonals, symmetry, and relationships between side lengths
- [Properties of rectangles](https://lightmysky.com/learn/mathematics/properties-of-rectangles-mt_KFKUR_gBg_): Use the properties of rectangles to deduce related facts and find missing lengths and angles
- [Angles that add up on a diagram](https://lightmysky.com/learn/mathematics/angles-that-add-up-on-a-diagram-mt_89riIKwGYp): Recognise angle measure as additive; find unknown angles by adding or subtracting on a diagram using equations with a symbol for the unknown
- [Understanding angles (age 10+)](https://lightmysky.com/learn/mathematics/understanding-angles-age-10-mt_QDTO3GAgcq): Recognise angles where they meet at a point, are on a straight line, or are vertically opposite; find missing angles using these properties
- [Understanding angles (age 11+)](https://lightmysky.com/learn/mathematics/understanding-angles-age-11-mt_tedML_iu4Y): Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems
- [Angles with parallel lines](https://lightmysky.com/learn/mathematics/angles-with-parallel-lines-mt_H_pNJ3ZI_S): Understand and use the relationship between parallel lines cut by a transversal: corresponding angles, alternate interior angles, and co-interior (same-side interior) angles; use these to find unknown angles
- [Coordinates (age 10+)](https://lightmysky.com/learn/mathematics/coordinates-age-10-mt_R4AY0LKxfl): Describe positions on the full coordinate grid (all four quadrants); use coordinates with negative values
- [Translating and reflecting shapes](https://lightmysky.com/learn/mathematics/translating-and-reflecting-shapes-mt_lxaM6iVpdr): Draw and translate simple shapes on the coordinate plane; reflect shapes in the axes
- [Area and volume formulae](https://lightmysky.com/learn/mathematics/area-and-volume-formulae-mt_tAJH5BrpOx): Derive and apply formulae for the area of triangles, parallelograms, and trapezia, and for the volume of cuboids and other prisms (including cylinders), connecting each formula to its geometric reasoning
- [Surface Area of Solids](https://lightmysky.com/learn/mathematics/surface-area-of-solids-mt_hdhmurUndc): Find the surface area of prisms and pyramids by unfolding them into nets of rectangles and triangles, working out each face's area and summing them; apply this to practical problems like wrapping boxes or painting walls
- [Circles: Circumference & Area](https://lightmysky.com/learn/mathematics/circles-circumference-and-area-mt_svFa6_mjO_): Calculate the circumference and area of circles using the formulae C = πd (or 2πr) and A = πr², and solve problems involving perimeters and areas of composite shapes that include circular parts
- [Volume and Surface Area of Prisms and Cylinders](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-prisms-and-cylinders-mt_JTLJ5LDrYh): Find the volume of any prism as base area times length, and its surface area as the two ends plus the wrapped rectangle, including the cylinder.
- [Angle sums in triangles and polygons](https://lightmysky.com/learn/mathematics/angle-sums-in-triangles-and-polygons-mt_J03RFlVdas): Derive and use the angle sum in a triangle (180°), use it to deduce the angle sum in any polygon ((n−2) × 180°), and calculate interior and exterior angles of regular polygons
- [Coordinate Transformations](https://lightmysky.com/learn/mathematics/coordinate-transformations-mt_K0Y15w48SY): Identify properties of translations, rotations, and reflections; describe and perform these transformations on given figures, and understand that the image is congruent to the original
- [Similarity and enlargement](https://lightmysky.com/learn/mathematics/similarity-and-enlargement-mt_y-BuQAfw4B): Understand similarity as a relationship where one shape is an enlargement of another; construct similar shapes by enlargement with a given scale factor and centre, with and without coordinate grids
- [Vectors: Notation, Addition and Scalar Multiples](https://lightmysky.com/learn/mathematics/vectors-notation-addition-and-scalar-multiples-mt_NkqmrL7pLe): Write vectors as columns or as bold letters, add and subtract them along a route, and multiply by a scalar to change length or reverse direction.
- [Accurate drawing and scale diagrams](https://lightmysky.com/learn/mathematics/accurate-drawing-and-scale-diagrams-mt_Jf8xcX4UTq): Draw and measure line segments and angles accurately using ruler and protractor, and interpret scale drawings to extract real measurements
- [Ruler and compass constructions](https://lightmysky.com/learn/mathematics/ruler-and-compass-constructions-mt_ZrsqGVG-Wt): Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle
- [Triangle congruence criteria](https://lightmysky.com/learn/mathematics/triangle-congruence-criteria-mt_167X6Ax8P7): Know and use the criteria for triangle congruence (SSS, SAS, ASA, RHS), use standard labelling conventions for sides and angles of triangle ABC, and determine whether two triangles are congruent
- [Bearings and Scale Drawings](https://lightmysky.com/learn/mathematics/bearings-and-scale-drawings-mt_9q32jC0Fk_): Describe a direction as a three-figure bearing measured clockwise from north, and use a scale drawing to move between a plan and real distances.
- [Congruent Triangles and the Congruence Conditions](https://lightmysky.com/learn/mathematics/congruent-triangles-and-the-congruence-conditions-mt_it8t7mY71J): Use SSS, SAS, ASA and RHS to decide whether two triangles are congruent, and write the decision as a short reasoned argument.
- [Similar Triangles and Proving Similarity](https://lightmysky.com/learn/mathematics/similar-triangles-and-proving-similarity-mt_NPRVlr4lrs): Decide whether two triangles are similar from equal angles or matching side ratios, set the reason out in one line, and use the scale factor to find a missing length.
- [Similar Shapes: Area and Volume Scale Factors](https://lightmysky.com/learn/mathematics/similar-shapes-area-and-volume-scale-factors-mt_v37c_AeLQa): Use the fact that areas scale by the square of the length scale factor and volumes by its cube, and work the relationship in both directions.
- [Exact Values of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/exact-values-of-sine-cosine-and-tangent-mt_SNJEbXYOrR): Know and use the exact values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees, with the acute values 30, 45 and 60 taken from a halved equilateral triangle and a half square, and the endpoints 0 and 90 read off the unit circle.
- [The Sine Rule](https://lightmysky.com/learn/mathematics/the-sine-rule-mt_cTt_-9KZho): Use a/sin A = b/sin B in triangles with no right angle to find a missing side or angle, and check whether an obtuse second answer also fits.
- [The Cosine Rule](https://lightmysky.com/learn/mathematics/the-cosine-rule-mt_yy73gK5CEP): Use a² = b² + c² - 2bc cos A when the sine rule does not apply: three sides given, or two sides with the angle between them.
- [Triangle Area with Half ab sin C](https://lightmysky.com/learn/mathematics/triangle-area-with-half-ab-sin-c-mt_PiwizT4D2b): Find the area of any triangle from two sides and the angle between them, and run the same rule backwards to find an angle from a known area.
- [Pythagoras' Theorem](https://lightmysky.com/learn/mathematics/pythagoras-theorem-mt_1VmTUxBrNd): Apply Pythagoras’ Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts
- [Pythagoras and Trigonometry in Three Dimensions](https://lightmysky.com/learn/mathematics/pythagoras-and-trigonometry-in-three-dimensions-mt_7yKxYqQHB2): Find lengths and angles inside cuboids, prisms and pyramids by picking a right-angled triangle out of the solid and solving it on its own.
- [Choosing a Triangle Method](https://lightmysky.com/learn/mathematics/choosing-a-triangle-method-mt_nubZzZjVvn): Decide from what a question gives you which tool a triangle needs: Pythagoras, right-angle ratios, the sine rule or the cosine rule.
- [Arc Length, Sector Area and Segments](https://lightmysky.com/learn/mathematics/arc-length-sector-area-and-segments-mt_z9LIwmfKYJ): Find arc lengths and sector areas as fractions of a whole circle, and find a segment by subtracting a triangle from its sector.
- [Bearing Problems with the Sine and Cosine Rules](https://lightmysky.com/learn/mathematics/bearing-problems-with-the-sine-and-cosine-rules-mt_YQFdU4aiFP): Turn a journey described by bearings into a triangle, then solve it with the sine or cosine rule instead of drawing it to scale.
- [Circle Theorems: Angles at the Centre and in a Semicircle](https://lightmysky.com/learn/mathematics/circle-theorems-angles-at-the-centre-and-in-a-semicircle-mt_kOwb61sfUR): Use the fact that the angle at the centre is twice the angle at the circumference on the same arc, and that the angle in a semicircle is a right angle.
- [Circle Theorems: Cyclic Quadrilaterals and the Same Segment](https://lightmysky.com/learn/mathematics/circle-theorems-cyclic-quadrilaterals-and-the-same-segment-mt_11GlmkiqB1): Use the equal angles subtended by the same arc, and the opposite angles of a cyclic quadrilateral adding to 180 degrees.
- [Tangents, Radii and the Alternate Segment Theorem](https://lightmysky.com/learn/mathematics/tangents-radii-and-the-alternate-segment-theorem-mt_Y-Xf0BOb1C): Use the right angle between a tangent and the radius at the point of contact, the equal tangents from an outside point, and the alternate segment theorem.
- [Volume and Surface Area of Cones, Spheres and Pyramids](https://lightmysky.com/learn/mathematics/volume-and-surface-area-of-cones-spheres-and-pyramids-mt_s4K5YMBSzG): Apply the standard formulae for pyramids, cones and spheres, and find a slant height when only the perpendicular height is given.
- [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4): Measure an angle by the arc it cuts on a circle of radius 1, so a full turn is 2π. In radians the arc length is rθ and the sector area is ½r²θ, with no fraction of 360 to carry.
- [Vectors in Three Dimensions](https://lightmysky.com/learn/mathematics/vectors-in-three-dimensions-mt_o97GclnNC-): Extend position, addition and scalar multiples to space, with distance and the standard unit vectors. Later topics in this sequence use this notation.
- [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp): Move between y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0, and produce the equation of a line from a point and a gradient or from two points.
- [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka): Use equal gradients for parallel lines and a gradient product of -1 for perpendicular ones, and apply both to find missing lines, midpoints and distances.
- [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe): Read a circle's centre and radius from (x - a)² + (y - b)² = r², and recover them from an expanded equation by completing the square in both variables.
- [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv): Find where a line cuts a circle by substitution, use the discriminant to tell a tangent from a chord or a miss, and use the right angle between a tangent and the radius.
- [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-): Define sine and cosine as the coordinates of a point travelling round the unit circle, which extends them beyond a right angle and explains the period, symmetry and repeated values of their graphs.
- [The Pythagorean and Quotient Identities](https://lightmysky.com/learn/mathematics/the-pythagorean-and-quotient-identities-mt_NXBq37vxQh): sin²θ + cos²θ = 1 is Pythagoras on the unit circle, and tan θ = sin θ / cos θ is a ratio of coordinates. Use both to simplify expressions and to prove further identities.
- [Parametric Equations of Curves](https://lightmysky.com/learn/mathematics/parametric-equations-of-curves-mt_7NC7SCeU2P): Describe a curve by giving x and y separately in terms of a parameter, plot it from a table of parameter values, and convert to Cartesian form by eliminating the parameter.
- [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA): Find every solution inside a stated interval, using the graph or the circle to get the solutions the calculator does not show, and adjusting the interval first when the angle is 2x or x + 30°.
- [Compound Angle Formulae](https://lightmysky.com/learn/mathematics/compound-angle-formulae-mt_ANDmX8exwg): Expand sin(A ± B), cos(A ± B) and tan(A ± B), and use them for exact values of angles built from known ones and for equations mixing two angles.
- [Double Angle Formulae](https://lightmysky.com/learn/mathematics/double-angle-formulae-mt_AC9MmZ_fam): Set B = A in the compound angle formulae to get sin 2A, cos 2A and tan 2A, and pick between the three forms of cos 2A to remove either a squared term or a doubled angle.
- [Combining a Sine and a Cosine into a Single Wave](https://lightmysky.com/learn/mathematics/combining-a-sine-and-a-cosine-into-a-single-wave-mt_IwQkNDRWyq): Write a sin x + b cos x as R sin(x + α), so two waves become one with a single amplitude and shift. Maximum and minimum values, and the x where they happen, then read off directly.
- [Conic Sections: Ellipse, Parabola and Hyperbola](https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu): Meet the three curves a plane cuts from a cone, in both their locus definitions and their standard equations, and read foci and axes off the equation.
- [Polar Coordinates and Polar Curves](https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v): Locate a point by distance and angle instead of two displacements, convert between the two systems, and sketch curves that are simpler in polar form.
- [Small Angle Approximations](https://lightmysky.com/learn/mathematics/small-angle-approximations-mt_fIsZes5Os3): Near zero, and only in radians, the sine of an angle is the angle, the tangent of it is the angle too, and the cosine drops off as one minus half the square. These are the approximations that turn an equation nobody can solve into one anybody can, and knowing when they stop being safe is half the point.
- [Proving Geometry Facts with Vectors](https://lightmysky.com/learn/mathematics/proving-geometry-facts-with-vectors-mt_A00MA7Ecgf): Show that one vector expression is a multiple of another, and use that to prove lines are parallel or that three points lie on a straight line.
- [Trigonometry basics](https://lightmysky.com/learn/mathematics/trigonometry-basics-mt_KB_Czd7RQH): Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem
- [The Dot Product: Angles and Projections](https://lightmysky.com/learn/mathematics/the-dot-product-angles-and-projections-mt_yFBcyE4sFw): One product of two vectors returns a number that encodes the angle between them. It tests perpendicularity and splits one vector into parts along and across another.
- [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK): The other product of two vectors returns a vector perpendicular to both, whose length is the area of the parallelogram they span and whose direction records the orientation.
- [Lines and Planes in Space](https://lightmysky.com/learn/mathematics/lines-and-planes-in-space-mt_kIX5ak1GRv): Write lines and planes in vector, parametric and Cartesian form, and answer distance and intersection questions with dot and cross products.
- [Arc Length and Curvature of a Space Curve](https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1): Reparametrise a curve by arc length and measure how fast its direction turns, separating the path from the speed along it.