On the Foundations Check, angles, shape and Pythagoras came back secure to step 5 and broke at step 6. Step 6 is the Extended rung. So the ordinary angle facts are in place — what is not yet in place is the Extended layer: circle theorems, similar-shape scale factors, and multi-step angle chases where one fact feeds the next.
Two other things from the check land directly on this topic:
180 − x and 360 − x. Most lost angle marks are not misunderstood theorems, they are
a subtraction done in a hurry.| Code | Sub-topic | Expected difficulty for you |
|---|---|---|
| E4.1 | Geometrical terms | Low — vocabulary, but Cambridge marks the words |
| E4.2 | Geometrical constructions | Low — method, not calculation |
| E4.3 | Scale drawings and bearings | Low — three-figure form catches people out |
| E4.4 | Similarity | Medium — the k² and k³ step is the dropped mark |
| E4.5 | Symmetry | Low |
| E4.6 | Angles | Medium — rebuilt here from the bottom |
| E4.7 | Circle theorems I | Medium — new territory above your break |
| E4.8 | Circle theorems II | Low, once the first five are secure |
Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. Every number in this guide is chosen so it can be done by hand, and surds are left as surds.
Number sense and the four operations: solid on all seven rungs. No gap anywhere, right up to Extended. Geometry is unusually kind to that strength — most circle-theorem questions are one clean subtraction or one halving, and you can already do those without thinking.
Which means the marks in this topic are not about arithmetic. They are about naming the reason. Cambridge awards the angle and the reason separately, and the reason is the mark most often thrown away by people who got the number right.
Every idea appears four times, with less help each time.
Section 9 is a mixed set: fifteen questions, unlabelled and out of order, because in an exam nobody tells you which theorem applies.
Geometry has more vocabulary than any other maths topic, and Cambridge tests the words directly. “Name the type of angle”, “state which two triangles are congruent”, “write down the mathematical name of the quadrilateral” — these are one-mark questions where the mark is the word.
| Word | What it means |
|---|---|
| Point | A position with no size. Labelled with a capital letter. |
| Line | Straight, and goes on for ever in both directions. |
| Line segment | The bit of a line between two points. AB usually means this. |
| Ray | Starts at a point and goes on for ever one way. |
| Plane | A flat surface going on for ever. A page is a piece of one. |
| Parallel | Same direction, never meet. Marked with matching arrowheads. |
| Perpendicular | Meeting at 90°. |
| Perpendicular bisector | The line that cuts a line segment exactly in half and crosses it at 90°. Every point on it is the same distance from both ends of the segment. |
| Vertex | A corner. Plural: vertices. |
| Word | What it is |
|---|---|
| Radius | Centre to the edge. Every radius in one circle is the same length — that fact does a lot of work later. |
| Diameter | Right across through the centre. Twice the radius. |
| Chord | A straight line joining two points on the circle. A chord through the centre is a diameter, the longest chord. |
| Tangent | A line that touches the circle at exactly one point. |
| Arc | Part of the curved edge. |
| Semicircle | Half a circle, cut off by a diameter. |
| Minor and major arc | Two points on a circle split it into two arcs: the shorter one is the minor arc, the longer one the major arc. Minor and major sectors and segments are named the same way. |
| Sector | A slice, bounded by two radii and an arc. |
| Segment | Cut off by a chord. The smaller piece is the minor segment. |
| Circumference | The whole way round the edge. |
| Shape | Properties Cambridge expects you to state |
|---|---|
| Equilateral triangle | Three equal sides, three angles of 60°. |
| Isosceles triangle | Two equal sides; the angles opposite them are equal. |
| Scalene triangle | All sides different, all angles different. |
| Square | Four equal sides, four right angles, diagonals equal and bisect at 90°. |
| Rectangle | Opposite sides equal, four right angles, diagonals equal. |
| Parallelogram | Two pairs of parallel sides, opposite angles equal, no line of symmetry. |
| Rhombus | A parallelogram with all four sides equal; diagonals cross at 90°. |
| Trapezium | Exactly one pair of parallel sides. |
| Kite | Two pairs of adjacent equal sides; one line of symmetry; diagonals cross at 90°. |
| polygon | sides | one interior angle if regular |
|---|---|---|
| pentagon | 5 | 108° |
| hexagon | 6 | 120° |
| octagon | 8 | 135° |
| decagon | 10 | 144° |
Regular means all the sides are equal and all the angles are equal. Anything else is irregular: a rectangle that is not a square has equal angles but unequal sides, so it is an irregular quadrilateral. One interior angle of a regular polygon is 180° − (360° ÷ number of sides).
| word | what it means |
|---|---|
| face | a flat surface of a solid |
| edge | where two faces meet |
| vertex (vertices) | a corner |
| prism | the same cross-section all the way along: cube, cuboid, triangular prism, cylinder |
| pyramid | a flat base and sloping triangular faces meeting at one point, the apex |
| hemisphere | half a sphere |
| frustum | a cone or pyramid with its top cut off parallel to the base |
0580 does not ask you to prove that two shapes are congruent, so there are no congruence rules to learn. What it does test is using the two words correctly. A reflection, rotation or translation always gives a congruent image; an enlargement gives a similar one, with every length multiplied by the scale factor.
E4.2 is three skills and only three: measuring and drawing lines and angles, constructing a triangle from its three sides with a ruler and a pair of compasses only, and drawing and using nets of solids. Many GCSE books and videos also teach bisector constructions and loci. The 0580 syllabus asks for neither, so they are not taught here.
Construction questions are marked on the method you can see. The arcs are the evidence. A perfect triangle with no arcs showing scores less than a slightly wobbly one with the arcs left in.
Four things must be on the desk: a sharp pencil, a ruler marked in mm, a protractor, and a pair of compasses that does not slip. Every straight edge is drawn with the ruler, never freehand.
Lines. Put the 0 mark of the ruler on one end of the line, not the end of the ruler — most rulers have a few millimetres of plastic before the 0. Read the other end to the nearest millimetre and write the length in centimetres to one decimal place (8.7 cm) or in millimetres (87 mm). 1 cm = 10 mm.
Angles. A protractor has two scales, one running each way round, so every line through its centre crosses two numbers that add up to 180. Only one of them is your angle.
Reflex angles. A reflex angle is between 180° and 360°, and a protractor stops at 180°. The two angles between the same pair of arms add up to 360°, so you work with the other one.
Accuracy. Measure and draw lengths to the nearest millimetre and angles to the nearest degree. Mark schemes allow only a small error either way, so draw thin, sharp lines and read each scale with your eye directly above it.
This is the syllabus's own example of a construction. All four sides of a rhombus are equal, so one diagonal splits it into two triangles whose three sides you already know. Draw the diagonal first, then do the triangle construction on both sides of it. This time you keep both crossing points: each one is the third vertex of a triangle.
A net is a flat shape that folds up into a solid. Every face of the solid appears on it exactly once, so the area of the net is the surface area of the solid. The net folds along the edges where two of its faces meet; every other edge on its outline is glued to another edge of the outline.
Six squares joined edge to edge can be arranged in 35 different shapes (turning a shape round or over does not make a new one). Exactly 11 of them fold into a cube, and these are all of them:
Three of the other 24, and why each one fails:
Opposite faces. In a straight line of three squares, the two end squares are opposite: the middle one folds up between them. The two end squares of a Z-shape of four are opposite too. In every cube net these two rules find at least two of the three pairs, and the two faces left over make the third pair. Two squares that share an edge on the net are never opposite.
Edges that meet. Where three squares sit round one point with a 90° gap, the two edges either side of the gap fold onto each other, so their far ends meet as well. That is how to answer a question asking which point meets which.
A cuboid is three pairs of equal rectangles. A triangular prism is two equal triangles and three rectangles, one rectangle for each side of the triangle. A square-based pyramid is a square with a triangle on each edge. Before adding anything up, count the faces against the solid: 6, 5 and 5.
Two skills sit in this sub-topic and they are usually examined together: reading a scale, and stating a bearing. Neither is difficult. Both are heavily penalised when written in the wrong form.
A scale of 1 : 25 000 means one unit on the map is 25 000 of the same unit in real life.
The units must match before you multiply.
040°, not 40°.| Direction | Bearing |
|---|---|
| North | 000° or 360° |
| North-east | 045° |
| East | 090° |
| South-east | 135° |
| South | 180° |
| South-west | 225° |
| West | 270° |
| North-west | 315° |
Two shapes are similar when one is an enlargement of the other: every angle equal, every pair of corresponding sides in the same ratio. That ratio is the scale factor, written k.
Get k the right way up by asking whether the answer should be bigger or smaller. Enlarging means k is greater than 1. If your k comes out less than 1 and the shape is clearly getting bigger, you have divided the wrong way round.
| What you are scaling | Multiply by | Why |
|---|---|---|
| Lengths, perimeters, radii, heights | k | One dimension |
| Areas, surface areas | k² | Two dimensions, each stretched by k |
| Volumes, capacities, masses of the same material | k³ | Three dimensions |
Two triangles are similar if two pairs of angles are equal; the third pair must then be equal too, because each triangle adds to 180°. A “show that” question wants each pair named with its reason: common angle, corresponding angles, alternate angles or vertically opposite angles. Then match the sides that sit opposite equal angles to find k.
Two kinds of symmetry, and they are counted differently. Line symmetry counts folds. Rotational symmetry counts how many times a shape fits onto itself in one full turn.
| Shape | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Square | 4 | 4 |
| Rectangle (not square) | 2 | 2 |
| Parallelogram | 0 | 2 |
| Rhombus | 2 | 2 |
| Kite | 1 | 1 |
| Trapezium (isosceles) | 1 | 1 |
| Equilateral triangle | 3 | 3 |
| Isosceles triangle | 1 | 1 |
| Scalene triangle | 0 | 1 |
| Regular polygon with n sides | n | n |
| Circle | infinitely many | infinite |
The angle you turn through is 360 ÷ order. Order 5 means a turn of 72°, and a shape that looks the same after 72° has order 5.
For a three-dimensional shape you count planes of symmetry — flat cuts that leave two mirror-image halves — and the order of rotational symmetry about an axis.
| Solid | Planes of symmetry | Rotational symmetry |
|---|---|---|
| Cuboid, all three edges different | 3 | order 2 about each of three axes |
| Cube | 9 | order 4 about each face axis |
| Cylinder | infinitely many, plus one across the middle | infinite about its main axis |
| Square-based pyramid | 4 | order 4 about the vertical axis |
| Sphere | infinitely many | infinite about any axis through the centre |
| Cone | infinitely many (every plane through its axis) | infinite about its axis |
| Prism whose cross-section is a regular n-sided polygon (not a cube) | n + 1: n through the main axis and 1 across the middle | order n about the main axis |
This is the sub-topic your check broke on, at step 6, so it is rebuilt here from the bottom rather than assumed. The facts themselves are step 3 and 4 material. What is Extended is chaining them — using one angle to unlock the next — and writing the reason every time.
Angle PRQ (also written ∠PRQ or PR̂Q) is the angle at R: the middle letter is always the vertex, and the other two letters are points on its two arms. Angle PRQ and angle QRP are the same angle. Exam questions name angles this way, so read the middle letter first.
Cambridge awards the angle and the reason on separate marks. The wording below is what mark schemes accept. Learn the phrases, not the idea — the idea you already have.
| The fact | The words that earn the mark |
|---|---|
| Straight line | angles on a straight line add to 180° |
| Full turn | angles at a point add to 360° |
| Crossing lines | vertically opposite angles are equal |
| Parallel lines, F shape | corresponding angles are equal |
| Parallel lines, Z shape | alternate angles are equal |
| Parallel lines, C shape | co-interior angles add to 180° |
| Triangle | angles in a triangle add to 180° |
| Triangle, side extended | the exterior angle of a triangle equals the sum of the two opposite interior angles |
| Isosceles triangle | base angles of an isosceles triangle are equal |
| Quadrilateral | angles in a quadrilateral add to 360° |
| Any polygon | exterior angles of a polygon add to 360° |
| Rule | Formula | Worth knowing |
|---|---|---|
| Angles in a triangle | 180° | Everything else is built on this |
| Angles in a quadrilateral | 360° | Two triangles stuck together |
| Interior angle sum, n sides | (n − 2) × 180° | n − 2 is the number of triangles it splits into |
| Exterior angle sum, any polygon | 360° | Does not depend on n at all |
| Regular polygon, one exterior angle | 360 ÷ n | Usually the fastest route in |
| Regular polygon, one interior angle | 180 − (360 ÷ n) | Interior and exterior are on a straight line |
Five theorems. They are the reason circle questions look impossible and then take two lines. Every one of them is a licence to write down an angle you were not given, and each has a set phrase that earns the reason mark.
| Theorem | The words that earn the mark |
|---|---|
| Angle in a semicircle | the angle in a semicircle is 90° |
| Tangent and radius | a tangent meets a radius at 90° |
| Angle at the centre | the angle at the centre is twice the angle at the circumference |
| Same segment | angles in the same segment are equal |
| Cyclic quadrilateral | opposite angles of a cyclic quadrilateral add to 180° |
One more that is not a theorem but does half the work in most questions: two radii always make an isosceles triangle, because all radii of a circle are equal. Reason: base angles of an isosceles triangle are equal.
Four more results. These are lower difficulty than section 7, but only once the first five are secure — they tend to appear as the second or third step of a longer question rather than on their own.
Drop a perpendicular from the centre onto a chord and it lands exactly on the midpoint. That manufactures a right-angled triangle whose hypotenuse is a radius, and Pythagoras finishes the job.
Two chords of the same length in one circle sit the same distance from the centre, and the converse holds: same distance means same length. It follows directly from the Pythagoras picture above — same radius and same distance forces the same half-chord.
From a point outside a circle you can draw exactly two tangents, and they have the same length. The four points make a kite: two radii equal, two tangents equal, two right angles where each tangent meets its radius. The line from the external point to the centre bisects the angle between the tangents.
That is Pythagoras again, using the right angle between tangent and radius.
Fifteen questions from everything above, shuffled and unlabelled. Deciding which fact applies is the part ordinary revision quietly does for you, and it is the part the exam tests. Before you calculate anything, say the reason out loud. If you cannot name a reason, you are guessing.
Six things, and between them they cover most of what an examiner can ask in topic 4:
Come back to section 9 in a week without reading anything above it. A high score on a cold run means this topic is set, and the next one can start.