IGCSE · Pearson Edexcel · 4MA1

Higher: The sine rule and ambiguous triangle data

Use named opposite side-angle pairs in the sine rule and retain every feasible triangle when SSA data have more than one branch.

Pearson 4MA1 · Higher Geometry · Trigonometry · Additional Higher 4.8C

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 triangle angle sums and positive side lengths.
Use acute and obtuse sine values and inverse functions.
Rearrange proportional equations with named quantities.

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

Quick readiness check

In standard triangle ABC notation, which pairing is valid in the sine rule?

What you will learn

Pair each lowercase side with its opposite capital angle.
Calculate general-triangle sides or angles from a known opposite pair.
Test both inverse-sine candidates against positivity, the angle sum and the supplied geometry.

In triangle ABC, A=42°, B=68° and a=BC=13 cm. Find C, b=CA and c=AB. Name the opposite pair used. 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.

Active geometry action

Given diagram

Whole view turn °

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

The two drawing axes use the same scale in cm. A whole-view turn rotates north and all geometry together. Free clicks preserve their actual coordinates without correction or snapping.

In triangle ABC, A=42°, B=68° and a=BC=13 cm. Find C,b=CA and c=AB. Name the opposite pair used.

Match each named angle to its opposite side

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My pair 1

Pair 1 angle vertex

Pair 1 opposite-side endpoint 1

Pair 1 opposite-side endpoint 2

My pair 2

Pair 2 angle vertex

Pair 2 opposite-side endpoint 1

Pair 2 opposite-side endpoint 2

My pair 3

Pair 3 angle vertex

Pair 3 opposite-side endpoint 1

Pair 3 opposite-side endpoint 2

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.

CMy response unit

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bMy response unit

Required for your current attempt. Give 3 significant figures.

cMy response unit

Required for your current attempt. Give 3 significant figures.

Equations and numerical substitutions for every requested part of Given diagram

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

Geometric conditions, units, branches and domain checks for Given diagram

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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.

Match every side with its opposite angle

In triangle ABC, a=BC lies opposite A, b=CA opposite B and c=AB opposite C. This naming convention is essential to the sine rule a/sin A=b/sin B=c/sin C. It is not a right-triangle ratio, and the two quantities in one fraction must genuinely be opposite each other.

With two angles known, first find the remaining angle from 180° and check it is positive. Use a supplied side and its known opposite angle as the reference pair. Rearrange the rule for each requested side and keep unrounded trigonometric values until final lengths are reported.

An inverse sine can leave two possible angles

When one known angle and two sides give sin B=r, a principal inverse may produce an acute candidate B₁. The supplementary value B₂=180°−B₁ has the same sine and may also fit a triangle. Test each separately rather than assuming the first calculator display is the only answer.

For every candidate, require positive side lengths and positive third angle C=180°−A−B. Then calculate any remaining side using that candidate’s own angle. Two admissible SSA branches have different angles and usually different third sides; drawing one triangle reflected across its baseline is not an additional SSA solution.

Recognise one triangle, no triangle and insufficient data

A useful SSA diagram places one side along a forward ray and locates the remaining vertex from the given side and angle. A circle with the required opposite-side radius can meet that ray twice, touch once or miss it. Only intersections satisfying the named givens and positive nondegenerate triangle conditions count.

If a proposed sine exceeds one, no such triangle exists. If a supplementary inverse makes A+B at least 180°, reject that branch; do not round it into a triangle. A right-angle inverse candidate is one candidate rather than two copies. These checks explain a complete answer instead of forcing every data set to have one solution.

Sources for this lesson

Pause and explain

A triangle proposal has A=30°, a=10 cm, b=15 cm. sin B=0.75 gives B≈48.6° or 131.4°. How many candidates survive the triangle angle-sum test?

Put the idea to work

Worked example

Show the worked solution

The known angles give C=180−42−68=70°. The given a=BC=13 cm is opposite A=42°.
Use b/sin 68°=13/sin 42°, so b=13 sin 68°/sin 42°≈18.0 cm.
Likewise c/sin 70°=13/sin 42° gives c≈18.3 cm. These three positive angles total 180°, and the longest side is opposite the largest angle 70°.

Answer C=70°, b≈18.0 cm and c≈18.3 cm. Every ratio uses an actual opposite side-angle pair.

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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

In triangle ABC, A=29°, B=83° and b=CA=17 cm. Find C, a and c, showing the known opposite pair. 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 known b, B opposite pair before writing ratios for a or c.

Compare my reasoning

C=180−29−83=68°. The supplied side is b=CA=17 cm, so the known opposite pair is b, B rather than a, A.
a/sin 29°=17/sin 83° gives a≈8.30 cm.
c/sin 68°=17/sin 83° gives c≈15.9 cm. The given b is longest, consistently opposite the largest angle 83°.

C=68°, a≈8.30 cm and c≈15.9 cm, using 17/sin 83° as the known reference ratio.

Look for these in your work

Pair each lowercase side with its opposite capital angle.
Calculate general-triangle sides or angles from a known opposite pair.
Test both inverse-sine candidates against positivity, the angle sum and the supplied geometry.

2 · Independent

Triangle ABC has A=35°, a=BC=9 cm and b=CA=12 cm, with no further location condition. Find every possible triangle, giving B, C and c. Explain why one calculator inverse is insufficient. 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

Keep both inverse-sine candidates and test each third angle before calculating its own third side.

Compare my reasoning

The opposite pair a=9, A=35° gives sin B=12 sin 35°/9≈0.7647686. This is between 0 and 1, so consider both inverse-sine angle candidates.
First B₁≈49.9° and C₁=180°−35°−B₁≈95.1°. Using the unrounded angle, c₁=9 sin C₁/sin 35°≈15.6 cm.
The supplement B₂=180°−B₁≈130.1° leaves C₂≈14.9°>0. Its own sine-rule calculation gives c₂≈4.03 cm.
Both candidates satisfy the given sides and positive angle sum, so retain two distinct triangles. They have different angle sets and third sides; a baseline reflection of one would not create a third SSA branch.

Two triangles: (B, C, c)≈(49.9°, 95.1°, 15.6 cm) or(130.1°, 14.9°, 4.03 cm). Both inverse-sine branches are admissible.

Look for these in your work

Pair each lowercase side with its opposite capital angle.
Calculate general-triangle sides or angles from a known opposite pair.
Test both inverse-sine candidates against positivity, the angle sum and the supplied geometry.

3 · Transfer

Compare three proposed triangles with A=30° and b=CA=14 cm, but a=BC equal to 7 cm, 6 cm or 16 cm. Determine every valid triangle for each proposal; give B, C and c when possible, with a geometric/domain reason for each count. 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

Compare each proposed opposite-side radius with the perpendicular height, then apply the positive remaining-angle test.

Compare my reasoning

With A=30° and b=14 cm, the perpendicular height from C to the forward AB ray is 14 sin 30°=7 cm. This controls how a circle of radius a about C can reach that ray.
For a=7 cm, sin B=1. The single candidate is B=90°, C=60° and c=7 sin 60°/sin 30°=7√3≈12.1 cm. The 90° candidate and its supplement coincide, so count one triangle.
For a=6 cm, sin B=7/6>1 and the circle cannot reach the ray. There is no triangle; do not force the input into the inverse-sine domain.
For a=16 cm, sin B=7/16. The acute candidate has B≈25.9°, C≈124.1° and c≈26.5 cm. Its supplement would make A+B>180°, so it is rejected.
The three proposals therefore have one, zero and one valid triangles respectively. Each count follows the actual side/angle conditions rather than an assumption that SSA always means two solutions.

a=7: one triangle, B=90°, C=60°, c=7√3≈12.1 cm. a=6: none. a=16: one, B≈25.9°, C≈124.1°, c≈26.5 cm.

Look for these in your work

Pair each lowercase side with its opposite capital angle.
Calculate general-triangle sides or angles from a known opposite pair.
Test both inverse-sine candidates against positivity, the angle sum and the supplied geometry.

Common mistakes

Pairing a side with a neighbouring rather than opposite angle.
Discarding an obtuse inverse-sine branch without testing the remaining angle.
Counting a reflected drawing as an extra SSA solution or clamping an impossible sine value.

Recall without your notes

What can you explain now?

Must every inverse-sine calculation in a triangle give two valid triangles?

Compare with the explanation

No. Test both angle candidates, but retain only distinct candidates with positive remaining angle and all named side conditions.

A supplement may make the angle sum at least 180°, a right-angle candidate may coincide with its supplement, or the given sine ratio may be impossible. Two inverse forms are candidates rather than an automatic triangle count.

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Make it stick

Write the sine rule with its opposite pairs, then explain how inverse-sine candidates can give two, one or no valid triangles.

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.