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IGCSE · Pearson Edexcel · 4MA1

Higher: Three-dimensional trigonometry and line-plane angles

Identify a line’s perpendicular projection onto a plane and use its actual right-triangle section to find the line-plane angle.

Pearson 4MA1 · Higher Geometry · Trigonometry · Additional Higher 4.8F
Exam scope and assessment notes

The first four lessons are Foundation content inherited by Higher; the last seven are additional Higher. Use the supplied right-angle, opposite-side, included-angle, north and perpendicular-foot conditions. Keep degree mode, physical units, positive lengths and all admissible triangle branches; round only final requested values. Line-plane angles are included; plane-plane angles are excluded. SSA branch and inverse-area comparisons are chosen safeguards, not extra source rows. Drawings and fixed comparisons support self-checking; numerical agreement and written attempts do not award marks, mastery, proof approval or instrument competence. Linear 4MA1 Issue 2 is checked against November 2026/June 2027 entries; regional options depend on centre eligibility and modular awards remain separate. Publisher links are navigation only. Book1 SS1–4 have literal tags; no Book2 tag or dedicated H4.8A Hub is invented. Public teaching-body evidence covers only Pythagoras pages74–76; later chapter bodies and Hub content are unverified, with SS2/SS3 pagination differences unresolved. Specialist approval and formal assessment remain pending.

Before you begin

  • Use three-dimensional Pythagoras and perpendicular sections.
  • Use acute ratios and inverse functions in degrees.
  • Identify a perpendicular foot, horizontal plane and signed vertical separation.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

For an oblique line AQ from A in a plane, Q projects perpendicularly to H in that plane. Which is the line-plane angle?

What you will learn

  • Locate the perpendicular foot on the stated plane and the line’s projection.
  • Distinguish a line-plane angle from the complementary angle to the plane’s normal.
  • Solve genuine 3D sections with unsigned line-plane angles and positive physical lengths.

A cuboid has base edges 9 m and 12 m and vertical height 8 m. From base corner A to opposite top corner G, find the line’s angle to the base plane. Name the perpendicular foot and projection, and find the line length. Give non-exact calculated lengths or areas to 3 significant figures and calculated angles to 0.1°, unless another precision is requested.

Choose your own geometry and enter every requested number, substitution and reason. Unlabelled lengths, angles and missing construction points are left for your calculation.

Given diagram

Given diagramBlue geometry and labelled values are givens. Amber traces retain my actual selected points and coordinates. Unlabelled angles and lengths are not reference answers.9 m12 m8 mABHDEFGK

Blue: original givens. Amber: my actual submitted geometry. Unlabelled sides and angles stay for calculation. The drawing can scroll horizontally on narrow screens.

This is a schematic 3D camera view: projected screen angles and distances are not metric measurements. Click an actual named point, or enter all three free coordinates explicitly; a screen click does not determine a unique 3D point. The tinted plane has an illustrative boundary.

A cuboid has base edges 9 m and 12 m and vertical height 8 m. From base corner A to opposite top corner G, find the line’s angle to the base plane. Name the perpendicular foot and projection, and find the line length.

Choose an actual visible named point on the board or from these buttons.

Identify or place the actual perpendicular foot on the plane

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My numbers, equations and reasons

Keep significant trailing zeroes and three-digit bearings in your own text. Changing a response or its unit closes any previous comparison or reference. Optional retained work stays visible until explicitly cleared.

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Written reasoning is recorded for human review; it is not automatically proved or marked.

Required for your current attempt.

Written reasoning is recorded for human review; it is not automatically proved or marked.

Required for your current attempt.

Written reasoning is recorded for human review; it is not automatically proved or marked.

This tool compares fixed mathematics. Written reasons need human review; it awards no marks, mastery, physical-instrument competence, specialist assessment or formal assessment approval.

Find the projection before choosing an angle

For a line from a point A in a plane to a point Q above or below it, drop the perpendicular from Q to the plane at H. The segment AH is the projection of AQ onto that plane. The smaller line-plane angle is the angle between AQ and AH at A, not an arbitrary angle that looks acute in a perspective sketch.

The perpendicular QH is a normal direction to the plane. In the oblique right-triangle section A-H-Q, the angle to that normal is complementary to the requested angle to the plane. Name the plane, foot and vertex before deciding which of those two different angles a question asks for.

Calculate the true projection and then the angle

A cuboid space line may project onto a base diagonal, so first calculate that diagonal from both base dimensions. Then use the vertical separation and the projection as the two legs in the right section. A face edge by itself is usually not the projection of a line joining opposite solid corners.

For an oblique line, tanθ=perpendicular separation/projection length; sine can instead use the true space-line length. In a right pyramid, use the given perpendicular apex-to-centre line and the centre-to-corner projection. Keep exact or unrounded intermediate lengths so the final angle reflects the actual solid.

Separate signed height from the smaller angle

A point below a horizontal plane has negative height relative to that plane, but perpendicular distance is the positive magnitude. The smaller angle between the line and plane remains between 0° and 90°. Above and below configurations with the same projection and separation magnitude therefore give the same unsigned line-plane angle.

A line parallel to the plane has angle 0°, while a perpendicular line has angle 90° even though its point projection collapses. Treat those boundary cases explicitly rather than inventing a nonzero projection for a ratio. Here the required geometry concerns a line and a plane; no angle between two planes is being requested.

Sources for this lesson

Pause and explain

An oblique line has perpendicular separation 5 m and plane-projection length 12 m. What is its smaller angle to the plane to 0.1°?

Put the idea to work

Worked example

A cuboid has base edges 9 m and 12 m and vertical height 8 m. From base corner A to opposite top corner G, find the line’s angle to the base plane. Name the perpendicular foot and projection, and find the line length. Give non-exact calculated lengths or areas to 3 significant figures and calculated angles to 0.1°, unless another precision is requested.

Show the worked solution
  1. Call H the base corner directly below G. GH is perpendicular to the base plane, and AH is the projection of the line AG onto that plane.
  2. The base projection satisfies AH²=9²+12²=225, so AH=15 m. The actual space line has AG²=15²+8²=289, so AG=17 m.
  3. The required angle is GAH, with tanθ=GH/AH=8/15. Hence θ≈28.1°. This is the angle to the base, not its complement to the normal.

Answer The foot H is the base corner directly below G; projection AH=15 m, AG=17 m and angle to the base≈28.1°.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

A vertical mast has foot O=(0, 0, 0) and top T=(0, 0, 9) m. A ground point A=(12, 5, 0) m is joined to T. Find the projection length, AT and the angle AT makes with the horizontal ground plane. Give non-exact calculated lengths or areas to 3 significant figures and calculated angles to 0.1°, unless another precision is requested.

Give me a hint

Identify the mast foot on the ground plane, then calculate the actual two-coordinate projection.

Compare my reasoning
  1. The top T projects perpendicularly onto the actual mast foot O in the ground plane. AO is the horizontal projection of AT.
  2. AO=√(12²+5²)=13 m, and AT=√(13²+9²)=√250=5√10 m≈15.8 m.
  3. The line-ground angle is TAO. tanθ=OT/AO=9/13 gives θ≈34.7°; using the vertical reference would instead describe a complementary angle.

Projection AO=13 m, AT=5√10 m≈15.8 m and angle to ground≈34.7°.

Look for these in your work

  • Locate the perpendicular foot on the stated plane and the line’s projection.
  • Distinguish a line-plane angle from the complementary angle to the plane’s normal.
  • Solve genuine 3D sections with unsigned line-plane angles and positive physical lengths.
2 · Independent

A right rectangular pyramid has base 12 m by 16 m, with apex S vertically 15 m above its centre. Find the angle between a sloping corner-to-apex edge and the base plane, and its length. Also state its complementary angle to the vertical. Give non-exact calculated lengths or areas to 3 significant figures and calculated angles to 0.1°, unless another precision is requested.

Give me a hint

Use the base centre as the supplied perpendicular foot and distinguish the requested base and vertical references.

Compare my reasoning
  1. Let A be the chosen base corner and O the base centre directly under the apex S. AO is the edge AS projection onto the base plane.
  2. The half-base legs are 6 m and 8 m, so AO=√(6²+8²)=10 m. The edge AS=√(10²+15²)=√325=5√13 m.
  3. The angle to the base is SAO=tan⁻¹(15/10)≈56.3°. The angle to the vertical is its complement≈33.7°, a separate angle with a different reference direction.

Projection=10 m; sloping edge=5√13 m≈18.0 m; angle to base≈56.3°, angle to vertical≈33.7°.

Look for these in your work

  • Locate the perpendicular foot on the stated plane and the line’s projection.
  • Distinguish a line-plane angle from the complementary angle to the plane’s normal.
  • Solve genuine 3D sections with unsigned line-plane angles and positive physical lengths.
3 · Transfer

For horizontal plane z=0, P=(3, −2, 0) m and Q=(11, 4, 10) m, identify the perpendicular foot of Q and find the line-plane angle and PQ. Repeat the angle conclusion for Q 2=(11, 4, −10), explaining why a below-plane endpoint does not create a negative line-plane angle. Give non-exact calculated lengths or areas to 3 significant figures and calculated angles to 0.1°, unless another precision is requested.

Give me a hint

Project to z=0 first and use the magnitude of vertical separation for the smaller line-plane angle.

Compare my reasoning
  1. The perpendicular foot is H=(11, 4, 0), obtained by retaining Q’s horizontal coordinates and placing it on z=0. The projected displacement PH is(8, 6, 0).
  2. PH=√(8²+6²)=10 m, while the perpendicular separation magnitude is 10 m. Thus tanθ=10/10 and the smaller line-plane angle is 45°.
  3. PQ=√(10²+10²)=10√2 m. For Q2=(11, 4, −10), the same foot and horizontal projection apply; the perpendicular-distance magnitude is still 10 m.
  4. The below-plane line therefore also makes 45° with the plane. A negative signed height does not create a negative smaller line-plane angle.

H=(11, 4, 0), PH=10 m, PQ=10√2 m≈14.1 m; both above- and below-plane line-plane angles are 45°.

Look for these in your work

  • Locate the perpendicular foot on the stated plane and the line’s projection.
  • Distinguish a line-plane angle from the complementary angle to the plane’s normal.
  • Solve genuine 3D sections with unsigned line-plane angles and positive physical lengths.

Common mistakes

  • Using a perspective-paper slope or one base edge instead of the actual plane projection.
  • Reporting the normal-angle complement when the question asks for the angle to the plane.
  • Making a below-plane line-plane angle negative or inventing a nonzero projection for a perpendicular line.
Recall without your notes

What can you explain now?

A line is perpendicular to a plane and its point projection collapses. Is its line-plane angle therefore unavailable or zero?

Compare with the explanation

No. A perpendicular line makes 90° with the plane; the collapsed projection is a boundary case, not a zero-angle line.

The oblique right-triangle tangent ratio needs a nonzero projection. The geometric perpendicular case is handled directly, while a line parallel to the plane has 0°; inventing a nonzero projection would change the line.

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Make it stick

Describe the actual foot and projected line used for a line-plane angle, contrast its normal-angle complement, and explain the parallel, perpendicular and below-plane cases.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics A

Check the official syllabus Issue 2 · November 2026 / June 2027 entries checked
Course scope and assessment

Linked teaching: sections 1.1 to 1.11, 2.1 to 2.8, 3.1 to 3.4,4.1–4.11 and5.1–5.2 and6.1–6.3; 2.5, 3.2,3.4 and5.1 are Higher only. This includes Foundation content inherited by Higher and Higher extensions for recurring decimals, surds, fractional powers, algebraic sets, Venn counts, repeated percentage changes, calculations with measurement bounds and standard-form application problems. Algebra includes Foundation symbols, integer index laws, substitution, like terms, brackets and monic factorisation, plus Higher fractional indices, multiple brackets, non-monic factorisation, algebraic fraction operations with domain restrictions, completing the square and proofs. Formulae include derivation from words and diagrams, substitution, single-occurrence subject changes including squares and roots, and Higher repeated-subject and harder power/root rearrangements. Linear equations cover both sides, brackets, fractional coefficients and constructing models from data. Section 2.5 adds all eight direct/inverse x, square, cube and square-root proportion families with actual graph exploration and domain checks. Section 2.6 adds shared elimination and substitution, exact fractions, contextual feasibility and repeated or inconsistent conditions; the additional Higher graphical lesson interprets genuine-line intersections, vertical lines and off-screen crossings. Section 2.7 adds Foundation monic quadratic factorisation, with Higher general factorisation, quadratic-formula and completing-square alternatives, context derivation and line–parabola/line–circle systems including full branches and exact paired solutions. Section 2.8 adds required open/closed number-line intervals, linear and compound solving, construction and identification of Cartesian regions, and Higher quadratic sign intervals and harder linear regions. Its Cartesian boundary-drawing convention note does not remove number-line endpoint requirements. Section 3.1 adds term-to-term and position-to-term generation, integer continuation and linear nth terms, plus Higher arithmetic a,d recovery, the general nth-term formula and finite arithmetic sums. Section 3.2 adds Higher-only mappings, both function notations, actual domain/range reasoning, composition in Pearson order with original restrictions, and affine/rational/restricted-branch inverses. Section 3.3 has nine shared coordinate, straight-line, travel, speed, conversion and quadratic-plotting lessons, followed by two additional Higher line-equation lessons, four Higher graph-family lessons, nine transformation/equation-recovery lessons and three drawn-tangent/intersection lessons. All nine Foundation source rows have lesson links, with paired interpretation and linear/quadratic plotting breadth guards. The broad additional Higher H3.3A row requires integer polynomial/cubic, sparse mixed reciprocal, degree sine/cosine and degree tangent lessons with every practice stage. H3.3B requires all four transformation forms across linear, quadratic, degree sine and degree cosine functions in eight core lessons; H3.3C also requires their algebra recovery and the ninth domain/equivalence/order synthesis lesson. H3.3D requires drawn-tangent construction and signed rise/run using actual units; calculus is not a replacement. H3.3E requires both line/quadratic and genuine line/cubic intersections and their nonlinear-minus-linear difference graphs, with all practice stages. All sixteen Graphs source rows now have teaching and staged-practice links; exhaustive subskill approval, mastery and assessment readiness remain unestablished. Section 3.4 adds six Higher-only lessons for variable rates, integer-power differentiation, gradient/rate conditions, stationary coordinates classified by original graph shape, one-dimensional motion and other practical models. H3.4C requires gradient and stationary slices; H3.4E requires both motion and other-practical slices. Preserve original domains, negative-power exclusions, admissible counts and boundary candidates. All five source rows have structural links, not exhaustive subskill approval or specialist/assessment validation. Section4.1 adds six shared Foundation/inherited-Higher lessons for selected acute/right/obtuse/reflex arcs, point/straight/intersection and parallel-transversal angles, triangle sums/exteriors, isosceles/equilateral and right/mixed triangles. All four shared rows have structural links; B requires both local and parallel strands and D both special and right-triangle strands, each with every practice stage. No additional H4.1 rows are invented. Preserve labelled geometry and reasons, the auxiliary-parallel triangle-sum argument, actual straight extensions, inclusive equilateral/isosceles and right-isosceles categories, and impossible/insufficient-data checks. Section4.5 constructions,4.4 measurement/bearings and the full H4.7 reasoning scope remain separate, without a blanket formal-proof waiver. Section4.2 adds seven shared Foundation/inherited-Higher Polygons lessons for all nine named polygon examples, convex/concave quadrilateral sums, all six named quadrilateral property families, general/regular polygon angles and congruence correspondence. Its seven rows A–G have structural links, with no additional Higher rows; C requires both family lessons and every practice stage. F/G remain two distinct published IDs supported by the same congruence lesson. Rhombus/square and kite endpoint-angle bisection and isosceles-trapezium equal diagonals are conservative standard-property breadth, not invented verbatim source bullets. Preserve reflex interiors, valid internal triangulations, local trapezium/kite conventions, exact integer side counts and impossible-data guards. Section4.3 symmetry,4.5 formal constructions,4.11 similarity and the full H4.7A reasoning programme remain separate. Literal book tags omit A; G has no Hub tag, and the Book2 map/proof congruence location and chapter7 page-range conflict remain unresolved. Section4.3 adds three shared Foundation/inherited-Higher Symmetry lessons for the one F4.3A source row, requiring every lesson and practice stage for the joint line/rotation capabilities. Draw all mirror lines, test a fixed original against a separate tracing about a chosen centre, count identity once and distinguish order1 from a blank circle’s non-finite rotations. Preserve indentations and true physical disk/colour features; the chosen missing-vertex parallelogram completion is original breadth, not an extra row or4.5construction credit. No additional H4.3 row is invented. The sole literal book link is Book2 Shape and Space6; no Book1 tag or explicit4.3Hub tag is invented, and its mapped chapter body/full final-book alignment remains unverified. Section4.5construction,5.2transformations,4.11similarity and full Higher reasoning remain separate. Section4.4 adds eight shared Foundation/inherited-Higher Measures lessons for seven A–G source rows and no additional Higher rows. Use complete instrument-scale, time-interval, sensible-estimate, nearest-degree protractor and three-figure-bearing teaching. F4.4F requires the full speed lesson; broad F4.4G requires speed, density and pressure, each with its worked explanation and every practice stage, with source binding on the same complete record. Preserve calibrated ticks and actual instrument action, compatible units, midnight/day order, approximate readings and north conventions. Keep numerical references hidden until calibrated attempts. Time recovery requires positive speed; at zero speed positive distance is impossible and zero distance alone is underdetermined. Density uses positive volume, displacement differences and explicit additive-volume assumptions; pressure uses the given formula, supplied forces and positive contact areas. Literal book navigation names only D and G; no A/B/C/E/F tags are invented, and Book1 map255–268/proof256–269 remains unresolved. These structural links do not complete4.5 instrument/compass constructions or scale drawings,5.2 transformations or formal assessment. Section4.5 adds six shared Foundation/inherited-Higher Construction lessons for four A–D rows and no additional Higher rows. A requires both nearest-millimetre measurement and actual line drawing. BroadB requires complete triangles and other two-dimensional shapes; broadD requires both straight-edge-and-compasses bisectors. Source binding and complete teaching/worked/all three practice stages must belong to the same required record. Use real ruler, protractor and compass action with retained construction evidence, physical calibration and honest approximate readings. Keep private endpoint readings, pose coordinates, solved vertices/crossings and reference directions hidden before instrument attempts. Respect valid mirror branches and selected minor/reflex angles; reject impossible triangles, insufficient data and degenerate crossings. Scale drawings require construction before measuring the unknown, compatible units and a labelled positive scale; drawing placement offsets remain givens. Literal book navigation names B/C/D only in Book2 Shape and Space6 with Hubs37/36/38. A has no literal chapter/Hub tag and no Book1 tag is invented. Public proof contents agree with map128–160 and show generic Constructions139, but the relevant body, full-book/Hub content and exact subtype pages remain unavailable. These structural links do not grant4.6 circle properties,4.7 whole geometrical reasoning,4.8 trigonometry,4.9 mensuration,4.11 maps/similarity,5.2 transformations or loci credit. The nearby circle formal-proof note is not imported. Section4.6 adds nine original Circle properties lessons for two shared Foundation and three additional Higher rows. The first three shared lessons cover all nine named parts, both chord/centre implications and both tangent facts. F4.6B requires both chord and tangent lessons. Additional Higher H4.6A requires both internal-chord and external-secant products with original named lengths, positive lengths and correct internal/near-far point order. H4.6B requires actual concyclic boundary vertices. H4.6C requires all five theorem parts across centre/circumference, diameter/same-segment, cyclic-opposite-angle and alternate-segment lessons. Every required slice needs substantive teaching, explained worked work and all three practice stages on one source-bound record; no half-row or untagged duplicate closes the bundle. Keep given marks separate from derived marks and retain selected major/minor arcs, tangent rays and physical algebraic-root domains. The no-formal-proof note belongs to H4.6C, not a blanket H4.6A/H4.7 exemption. Literal Book2 chapter5 tags F4.6B/H4.6B/H4.6C and chapter8 tags H4.6A/H4.6C are navigation only; no F4.6A or Book1 tag is invented. Chapter8 map326–349 versus proof chapter-start325 and next chapter349 remains unresolved; relevant final bodies and Hub content are unverified. These structural links do not complete construction, whole geometrical reasoning, trigonometry or mensuration, or establish specialist approval, mastery or formal assessment. Section4.7 adds five original Geometrical reasoning lessons for one shared F4.7A and one additional H4.7A row. The shared row jointly requires line/triangle, polygon and chord/tangent numerical reasoning, including lengths; the Higher row jointly requires both circle and mixed-angle standard-statement chains. Every required record needs substantive teaching, its explained worked example and all three fresh practice stages, with the source ID on that same complete record. Preserve actual diagram hypotheses, common arcs, contact radii, selected tangent rays, cyclic/straight point order, valid internal triangulations and the unknown regular-polygon outline after deduction. Each geometric equation needs a named reason; a correct number or sufficient text length does not validate an arbitrary written argument. The H4.6C formal-theorem-proof note remains local, not a blanket4.7 waiver. Only literal H4.7A Book1 Shape and Space1/Hub89,137,138,139 navigation is used; no Foundation4.7A or Book2 tag is invented. Public Book1 chapter60–99 is available but does not establish full Higher circle-angle or Hub body scope. These links do not award4.5construction skill, specialist approval, mastery or formal assessment. Section4.8 adds eleven original Trigonometry and Pythagoras’ theorem lessons for three shared F4.8A–C and six additional H4.8A–F rows. The first four are shared; the last seven are additional Higher. F4.8B jointly requires lengths with all acute ratios and inverse-angle work; H4.8C jointly requires sine-rule and cosine-rule teaching. Every row needs all its named source-bound records, substantive teaching, explained worked work and every Guided/Independent/Transfer stage. Preserve both missing-leg and missing-hypotenuse Pythagoras, signed bearing components with local north, all three obtuse functions, actual elevation/depression horizontals, admissible SSA branches and included-angle area restrictions. Three-dimensional work requires perpendicular sections and the actual foot/projection for a line-plane angle; plane-plane angles are excluded. Positive physical lengths, inverse domains, triangle inequalities, degree mode, exact forms and final-only rounding remain explicit. Chosen SSA two/one/no and inverse-area comparisons are safeguards, not extra syllabus rows. Literal navigation uses Book1 SS1–4 only: no Book2 tag or dedicated H4.8A Hub is invented. Hub145 is tagged F4.8A in SS1 and H4.8D in SS3; the former title does not extend Foundation into3D. Public Pythagoras body74–76 is available, but SS2/3/4 bodies and Hub content are unverified. SS2 map176–197 versus summary198, and SS3 map243–254 versus proof-start244/summary255 remain unresolved. No4.5construction,4.9mensuration, full-book, specialist, mastery or formal-assessment credit follows from these structural links. Section4.9 adds eight original Mensuration lessons for five shared F4.9A–E and one additional H4.9A row. The first six are shared; the last two are Higher. F4.9E jointly requires complete circle and semicircle/compound-boundary lessons; H4.9A jointly requires degree arcs/perimeters and sector-area/recovery lessons. Every required source-bound record needs substantive teaching, its explained worked example, checks and all Guided/Independent/Transfer stages. Retain both-direction linear/area conversions, exposed polygon boundary traces and internal joins, triangle/rectangle decomposition and contained disjoint cutouts, outside perpendicular heights, both parallelogram/trapezium methods, radius/diameter recovery and semicircle arc-versus-perimeter distinctions. Preserve major/remaining sector sweeps, positive physical roots, inverse dimensions, annular inner arcs and radial gaps, compatible square/linear units and question-specific precision; no universal3sf rule is invented. Radians are excluded; ordinary sector arc-plus-two-radii boundaries require0<θ<360, while a full disk has no radial cuts. Literal Book1 SS1 tags B/C/D/E and HigherA with Hubs47/48/118, plus Number4A/Hub14, are navigation only; no Book2 tag or verified Hub body is inferred. Public SS1contents summary98 versus actual99 and Number4map198–206 versus proof199–207/next208 stay explicit. Chosen compound/inverse/annular safeguards do not add source rows or grant construction, volume, similarity, full-book, specialist, mastery or formal-assessment credit. Section4.10 adds nine original 3D shapes and volume lessons for six shared F4.10A–F rows and one additional H4.10A row. The first seven are shared; the final sphere and right-cone lessons are additional Higher. Teach all seven named solid families, complete face/edge/vertex incidence with hidden features, planar triangle/rectangle surface inventories and cylinder surfaces with actual open/closed conditions. F4.10E requires both complete prism and cylinder-volume slices; H4.10A requires both complete sphere and right-cone surface-and-volume slices. Every required source-bound record needs substantive teaching, worked methods, checks and all Guided/Independent/Transfer stages. Preserve actual face altitudes, positive inverse dimensions, full-length groove conditions, internal capacity and fill levels, contact-patch removal, cubic unit factors and1L=1000cm³. Curved-solid tasks state their flat-face/rim/apex convention without inventing a universal mark-scheme count. Nets, joins, inverse applications and solid hemispheres are chosen breadth, not extra official rows or instrument, frustum, similarity, pyramid-volume or hollow-capacity credit. Keep exact values/π and question-specific precision, including explicit significant trailing zeroes or scientific form. Literal Book2 Shape and Space9 tags C/D/E/F and HigherA are navigation only; no A/B or Book1 tags are inferred. Hubs56/57/55/84/131 do not establish body coverage, and no F4.10F Hub tag is invented. Map359–385 versus proof358–384/next385 remains unresolved; relevant chapter body and full-book/Hub content are unavailable. Specialist, formal-assessment and mastery approval remain pending. Section4.11 adds seven original Similarity lessons for exactly two shared F4.11A/B rows and three additional H4.11A/B/C rows. The first four are shared correspondence, forward/inverse lengths, maps and plans; the final three cover area factors, volume factors and mixed problems. F4.11A requires both complete correspondence and length slices, and F4.11B requires both map and plan slices, with source binding, substantive teaching, worked methods, checks and every explained Guided/Independent/Transfer stage in the same actual record. Retain matching angles and every corresponding side, concave reflex corners, positive factors, inverse recovery, parallel/shared-vertex hypotheses, whole/part order, compatible units, route versus direct distance, and uniform print-resizing scale recovery. Higher work distinguishes k, k² and k³, positive roots, similarity hypotheses, unit powers, actual closed-cuboid total surfaces, physical dimensions, exact money before rounding, and parallel-section cone subtraction as a chosen application. No extra official row, transformations, instrument calibration, formal construction or general frustum-formula credit is claimed. Literal Book2 Shape and Space6 F4.11B/Hub36, Shape and Space7 F4.11A/Hub85, and Shape and Space9 H4.11A/B/C with Hubs143/144 are navigation only, with no Book1 tag invented. Shape and Space6 map128–160/proof128/summary159/next161, Shape and Space7 map223–248/proof222/summary247/next248, and Shape and Space9 map359–385/proof358/summary384/next385 remain explicit. The actual proof places Similar shapes at147 under Shape and Space6 while the literal F4.11A map tag remains under Shape and Space7; this topic-location conflict is unresolved. Relevant chapter and Hub bodies are unavailable; no full-book, specialist, formal-assessment or mastery approval follows. Section5.1 adds nine original additional Higher Vectors lessons for exactly H5.1A–G and no shared Foundation rows. All required source-bound notation, scalar, arithmetic, magnitude, route, position/division, parallel/collinear, simple-proof and intersection strands need substantive original teaching, an explained worked example and exactly one complete Guided/Independent/Transfer task in the same record. Preserve directed endpoints, signed/scalar and zero-vector exceptions, exact fractions and radicals, final-only rounding, current own head-to-tail joins, one-origin positions, t=0/t=1 endpoints and external points without clamping. Geometric conclusions require actual nonzero, nonparallel, noncollinear or simple-boundary hypotheses; formal two-to-four-symbol planar identities do not assert an independent spatial basis. H5.1G requires genuine explained geometric proof, not numerical resemblance or unchecked text. The H4.6C local theorem-proof note is not a blanket vector-proof waiver. Literal Book2 Shape and Space10 H5.1A–G map427–450/proof426/summary449/next450 remains unresolved; Hub167 covers A/B/C/D/F,168 A/E and169 G. Navigation does not establish modular equivalence. No Book1 tag or unseen Vectors chapter/Hub body is invented: the public uncorrected2025proof exposes contents and Number8 body, not Vectors. November2026/June2027 linear entry and regional-centre limits remain separate. Full-book, specialist, mastery and formal-assessment approval remain pending. Section5.2 adds twelve original shared Transformation geometry lessons across exactly F5.2A–M and no additional Higher row. Full source-bound bundles require substantive teaching, explained worked and exactly one complete Guided/Independent/Transfer task in each record. Retain every labelled corner and corresponding trace, signed/general rotations, complete equivalent mirror lines, common translation columns, rigid distances/angles, positive and fractional factors, supplied/free centres and factor1 identity exceptions. General-degree written3dp is separate from unrounded geometry. Ordered compositions are chosen breadth rather than an extra source row; compare both orders, including commuting perpendicular reflections, and distinguish reverse order from the inverse. Source/book/year and proof/physical-construction limits remain explicit. Graphical representation6.1 adds ten original lessons: six shared Foundation tabulation/two-way/pictogram/bar/pie/interpretation slices and four additional Higher histogram construction/reading and cumulative construction/use slices. Each complete source-bound record and every practice stage are required; tables or diagram names alone do not establish actual construction. Keep scales/units/keys, zero categories, unequal continuous widths and density/area, hidden missing calibration and pie labels, uniform partial-class estimates, cumulative boundaries and nonunique flat inverses. No Statistical measures6.2 or Probability6.3 credit follows. Literal book/Hub metadata is navigation only: Book2 Handling Data4 F6.1Aii is a modular subpart; Book1 Handling Data3 map398 starts differently from proof397 and Book2 Handling Data6 map208 from proof207. Relevant chapter/Hub bodies remain unavailable. Statistical measures6.2 adds eight original lessons: four shared Foundation list/frequency averages, simple mean problems and grouped midpoint estimates/modal classes; four additional Higher spread, discrete quartiles, median and IQR from cumulative diagrams. Every source-bound record needs substantive teaching, its explained worked example and all three unique practice stages. Preserve repeated positions, weighted counts, observed zero-frequency extremes, compatible units, grouped estimation limits, the explicitly chosen median-of-halves convention, actual graph guides, each curve’s own total and nonunique plateau intervals. Literal Book2 Handling Data4/5/6 tags and Hubs are navigation metadata only; no Book1 tag, complete chapter body or Hub access is inferred. Handling Data6 map208–222 versus proof207/summary221/next222 stays explicit. Statistical measures alone does not replace Probability6.3 teaching. Probability6.3 adds nine original lessons for ten shared F6.3A–J and four additional H6.3A–D rows. The first five are shared language/scales/models, complete single/two-event sample spaces, Venn probabilities, experimental estimates/expected frequencies and complement/mutually-exclusive addition; the final four are Higher independent trees, actual without-replacement trees, simple conditioning and contextual inverse/mixed models. Each required source-bound strand needs substantive teaching, an explained worked example and exactly one complete Guided/Independent/Transfer task in the same actual record. F6.3C requires both model and sample-space slices; H6.3A requires independent, dependent and application trees; H6.3B requires independent and application slices; H6.3C requires without-replacement, simple-conditioning and application slices; H6.3D requires conditioning and application slices. Retain physical identities, nonuniform weights, original ordered histories, complete branches/terminal events, unchanged versus changing contents, restricted positive conditional denominators and undefined empty conditioning. Experimental variability and fractional expected counts are model predictions rather than guaranteed observations. Inverse physical counts keep original nonnegative integer/positive-total domains. Chosen three-set, three-stage and inverse safeguards do not add source rows or a named general conditional theorem. Literal Book1 HandlingData1/2/3 tags and Hubs are navigation only; no Book2 tag or unseen probability chapter/Hub body is invented. Preserve map228–242/proof229–243/next244, map303/proof304–328/next329 and map398–428/proof397/summary426/next428; missing explicit Hub A/B/F/J tags remain absent. The common two-coin elementary model is not claimed as novel. All39 source sections and242 requirements now have structural teaching/practice links; that is not exhaustive subskill, teacher, full-book, mastery, specialist or formal-assessment approval. The independent source inventory contains 162 Foundation requirements and 80 additional Higher requirements; some lettered requirements contain multiple skills. A lesson link alone does not verify every skill or examination readiness. Written practice is self-checked, not automatically marked. Specialist approval and cumulative assessment have not been established. This course follows the linked Issue 2 linear 4MA1 specification. The 2026/2027 entry manual lists F/H in November 2026 and June 2027; regional FR/HR entries are June 2027 only and depend on centre eligibility. Modular 4XMAF and 4XMAH coverage is separate.

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