G Geometry and measures — notes on one page

G1 · Use conventional terms/notations: points, lines, vertices, edges, planes, parallel and perpendicular lines, right angles, polygons; standard triangle labelling; draw diagrams from written description

• Geometry uses agreed words and symbols so a diagram communicates facts precisely. A point names a position; a line continues, while a line segment has endpoints.
• Vertices are corners and edges join vertices. Parallel lines never meet, whereas perpendicular lines meet at $90^\circ$.
• In $\angle ABC$, the middle letter $B$ is the vertex; in triangle $ABC$, side $AC$ is opposite $B$.
• Draw a written description one instruction at a time, label every required point and add conventional arrow, equality and right-angle marks.
• The examiner awards stated or marked relationships, never what a sketch merely appears to show.

Exam tip. In a construction or drawing question, keep every requested label and conventional mark visible because each can carry an independent mark.

G2 · Use standard ruler and compass constructions (perpendicular bisector, perpendicular from/at a point, angle bisector); construct figures, solve loci problems; perpendicular distance is shortest

• Ruler-and-compass constructions must show the arcs that create the result. For a perpendicular bisector of $AB$, draw equal-radius arcs from $A$ and $B$ with radius greater than half of $AB$, then join their intersections.
• Every point on this line is equidistant from $A$ and $B$.
• An angle bisector is built from an arc centred at the vertex and equal arcs from the two cut points.
• Translate loci language into boundaries: a fixed distance from a point gives a circle; a fixed distance from a line gives two parallels.
• Perpendicular distance is the shortest distance to a line.

Exam tip. For a loci question, draw each condition separately, then shade or state only their intersection.

G3 · Apply angles at a point, on a straight line, vertically opposite angles; use alternate and corresponding angles on parallel lines; derive and use the angle sum of a triangle and of any polygon

Definition. Specification G3: apply the properties of angles at a point, angles at a point on a straight line and vertically opposite angles; understand and use alternate and corresponding angles on parallel lines; derive and use the sum of angles in a triangle, and use it to deduce the angle sum in any polygon and the properties of regular polygons.

• Angles at a point total $360^\circ$, angles on a straight line total $180^\circ$, and vertically opposite angles are equal. Where a line crosses two parallel lines, corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles add up to $180^\circ$.
• Angles in a triangle add up to $180^\circ$, and an exterior angle of a triangle equals the sum of the two interior opposite angles.
• Splitting an $n$-sided polygon into $n-2$ triangles gives an interior-angle sum of $(n-2)\times180^\circ$; the exterior angles of any polygon total $360^\circ$, so a regular polygon has exterior angle $360^\circ\div n$.
• In a reasoning question each step needs a calculation and the reason for it, with the key words in the reason and the reason tied to the parallel lines or triangle actually used.
• Angles found should be marked on the diagram or named.

What the examiner wants. The correct name for the fact used. The November 2024 Paper 1H report says the reasoning mark was for an appropriate parallel-lines reason, usually “alternate angles”, and that “corresponding angles” was a common wrong reason for the same angle; the June 2024 Paper 2H report warns that showing the angles sum to $360^\circ$ does not show a shape is a trapezium.

Exam tip. A 'give a reason' angle question needs the named fact, such as 'alternate angles, parallel lines', not just the arithmetic.

G4 · Derive and apply properties and definitions of special quadrilaterals (square, rectangle, parallelogram, trapezium, kite, rhombus), triangles and other plane figures using appropriate language

• Classify a plane shape using properties that must hold, not how the drawing looks. A parallelogram has two pairs of parallel opposite sides; its opposite sides and angles are equal, and its diagonals bisect each other.
• A rectangle is a parallelogram with four right angles and equal diagonals. A rhombus has four equal sides and perpendicular diagonals, while a square is both a rectangle and a rhombus.
• A kite has two pairs of adjacent equal sides.
• An isosceles triangle has equal base angles.
• Examiners expect a property to be connected to a conclusion, especially in 'explain' and proof questions.

Exam tip. For a classification question, give the most specific shape forced by the information and state the defining property.

G5 · Use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS)

• Congruent triangles have exactly the same size and shape, so corresponding sides and angles are equal. Prove congruence using SSS, SAS, ASA or RHS.
• SAS needs the angle included between the two known sides. RHS applies only to right-angled triangles and needs equal hypotenuses plus one equal shorter side.
• Mark or state the three matching facts, then write triangle names in corresponding order.
• Only after proving congruence may you use corresponding parts to establish another equality.
• AAA proves similarity rather than congruence, and SSA does not determine a unique triangle, so neither is a valid general congruence test.

Exam tip. In a proof, state the three matching facts before naming the congruence criterion and the corresponding conclusion.

G6 · Apply angle facts, congruence, similarity and quadrilateral properties to derive results about angles and sides, incl. Pythagoras' theorem and isosceles base angles, and obtain simple proofs

• Combine established geometry facts to derive a new result. Equal sides in an isosceles triangle face equal angles, and equal angles face equal sides.
• In a right-angled triangle, Pythagoras' theorem gives $a^2+b^2=c^2$, where $c$ is always the hypotenuse opposite the right angle.
• Angle facts, quadrilateral properties, similarity and congruence can then form a proof chain.
• Each line must follow from given information or a named result; a diagram's appearance is not evidence.
• For a 'prove' question, show the intermediate equality or congruence and finish with the exact conclusion requested.

Exam tip. A simple proof needs a reason beside each claim; a correct final statement without the linked reasons does not earn all proof marks.

G7 · Identify, describe and construct congruent and similar shapes, incl. on coordinate axes, by rotation, reflection, translation and enlargement (including fractional and negative scale factors)

• Describe a transformation completely: a translation needs a column vector; a rotation needs centre, angle and direction; a reflection needs its mirror line; an enlargement needs centre and scale factor. Translation, rotation and reflection preserve every length and angle, so object and image are congruent.
• Enlargement preserves angles and multiplies every length from the centre by the scale factor. For centre $C$, use $\overrightarrow{CP'}=k\overrightarrow{CP}$.
• A fractional factor shrinks the shape.
• At Higher tier, a negative factor also puts every image point on the opposite side of the centre.
• Examiners require every defining detail, not just the transformation name.

Exam tip. For an enlargement, draw straight rays from the centre through corresponding points to check the centre and scale factor.

G8 · Describe the changes and invariance achieved by combinations of rotations, reflections and translations [Higher only]

• Higher tier only. In a combination, apply transformations in the stated order because the image from one step becomes the object for the next.
• Two translations combine by adding their vectors. Two reflections in parallel lines produce a translation; two reflections in intersecting lines produce a rotation, with angle twice the angle between the mirror lines.
• Rotations, reflections and translations preserve lengths, angles, area and parallelism. One reflection reverses orientation; two reflections restore it.
• Coordinate rules are a reliable way to identify a single equivalent transformation.
• Reversing the order can change the result, so transformations do not generally commute.

Exam tip. To identify a combined transformation, track one general point and then state every parameter of the single result.

G9 · Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment

• Use circle vocabulary precisely. A radius joins the centre to the circumference.
• A diameter is a chord through the centre and has length $2r$. A chord joins two points on the circumference, while a tangent meets the circle at exactly one point.
• An arc is part of the circumference. A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
• The minor region uses the shorter arc and the major region uses the longer arc.
• In a labelled-diagram question, identify the defining boundaries rather than relying on a region's appearance.

Exam tip. For a circle-definition mark, name the boundary pieces explicitly: radii, chord or arc.

G10 · Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results [Higher only]

• Higher tier only.
• Standard circle theorems connect angles, radii, tangents and chords.
• The angle at the centre is twice the angle at the circumference standing on the same arc; angles in the same segment are equal; an angle in a semicircle is $90^\circ$; and opposite angles in a cyclic quadrilateral total $180^\circ$.
• A radius is perpendicular to a tangent at the contact point, tangents from one external point are equal, and the alternate segment theorem equates the tangent-chord angle with the angle in the opposite segment.
• A proof must identify the relevant common arc or chord and name each theorem used.

Exam tip. A circle-theorem 'give a reason' mark needs the theorem's name or an unambiguous full statement.

G11 · Solve geometrical problems on coordinate axes

• Coordinate geometry turns a diagram into exact calculations. The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Horizontal and vertical coordinate changes form a right triangle, so distance is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
• Coordinate differences can prove equal side lengths and identify horizontal or vertical lines; areas can be found from base and perpendicular height or by enclosing and subtracting simpler shapes.
• Keep brackets when subtracting negative coordinates and show the substitution.
• The axes and calculated differences, not the sketch's appearance, supply the evidence the examiner needs.

Exam tip. In a 'show that' coordinate proof, display the gradients or squared lengths that establish the claimed property.

G12 · Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

• Describe a solid by its flat faces, curved surfaces, edges and vertices. A face is flat; an edge is where surfaces meet; a vertex is where edges meet.
• An $n$-sided prism has two congruent polygonal ends and $n$ rectangular side faces, so it has $n+2$ faces, $3n$ edges and $2n$ vertices.
• A pyramid has one polygonal base and triangular faces meeting at an apex.
• A cylinder has two circular faces and one curved surface but no vertices; a cone has one circular face, one curved surface and one vertex; a sphere has only one curved surface.
• Count hidden features once.

Exam tip. For a prism, identify the number of sides on one end first, then use $n+2$, $3n$ and $2n$ as a check.

G13 · Construct and interpret plans and elevations of 3D shapes

• A plan is the view from directly above. A front or side elevation is an orthographic view from the named horizontal direction, drawn without perspective.
• Corresponding widths align between the plan and front elevation; depths align between the plan and side elevation. For stacks of unit cubes, a numbered plan records the height in each occupied cell.
• An elevation shows, for each viewing column, the greatest height visible along the line of sight.
• Use true lengths in the viewing plane and include changes of outline, but do not add hidden internal edges unless requested.
• Examiners compare dimensions and visible column heights, not artistic realism.

Exam tip. Before drawing an elevation, write the two dimensions visible from that direction and project matching corners with straight construction lines.

G14 · Use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money, etc.)

• Choose standard units that match the quantity: length uses mm, cm, m or km; area uses square units; volume uses cubic units; capacity uses ml or litres; mass uses g or kg; and time uses seconds, minutes or hours.
• Convert all measurements to compatible units before calculating.
• A linear conversion factor must be squared for area and cubed for volume: because $1\text{ m}=100\text{ cm}$, $1\text{ m}^2=10\,000\text{ cm}^2$ and $1\text{ m}^3=1\,000\,000\text{ cm}^3$.
• Examiners expect a numerical value with the correct unit and sensible rounding for the context, especially money and time.

Exam tip. Write the unit conversion before the arithmetic; this makes the required power and final unit visible for method marks.

G15 · Measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

• Measure only from an accurate scale drawing, using a ruler or protractor to a precision justified by the scale. For scale $1:n$, one unit on the drawing represents $n$ of the same units in reality, so multiply drawing lengths by $n$ and then convert units.
• A bearing is the clockwise angle from north at the starting point and is written with three figures, such as $052^\circ$.
• Draw a north line at the point where the journey starts.
• Reverse bearings differ by $180^\circ$.
• Examiners expect the scale working, the correct direction of measurement and a three-figure bearing.

Exam tip. For a bearing mark, draw the north line at the departure point and write the answer using exactly three figures.

G16 · Know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of cuboids and other right prisms (including cylinders)

Definition. Specification G16: know and apply formulae to calculate the area of triangles, parallelograms and trapezia, and the volume of cuboids and other right prisms, including cylinders.

• Area formulae: triangle $\frac12bh$, parallelogram $bh$, trapezium $\frac12(a+b)h$, where $h$ is the perpendicular height between the base and the parallel side, never a sloping edge. A right prism has the same cross-section all the way along, so $V=\text{area of cross-section}\times\text{length}$; a cylinder is the case with a circular cross-section, $V=\pi r^2h$.
• Identify and draw the cross-section first, find its area in square units, then multiply by the length to get cubic units.
• Compound shapes are split into rectangles and triangles, or a larger shape with a piece subtracted; each area is a process mark.
• Convert to consistent units before dividing one area by another, for example tiles into a wall.
• Units are often a separate mark, so write cm$^2$ or cm$^3$ with the number.

What the examiner wants. The cross-section identified before any multiplying. The June 2024 Paper 3H report says many could not find the volume of a cylinder, with errors ranging from finding the surface area to multiplying the radius by the height, while the Paper 1H report notes that splitting the compound floor into two rectangles and showing each area earned the process marks even when the arithmetic slipped.

Exam tip. For a prism, write 'cross-sectional area × length' before substituting so both stages of the method are clear.

G17 · Know circumference of a circle = 2πr = πd and area = πr²; calculate perimeters of 2D shapes incl. circles, areas of circles and composite shapes, surface area and volume of spheres, pyramids, cones

Definition. Specification G17: know the formulae circumference of a circle $=2\pi r=\pi d$ and area of a circle $=\pi r^2$; calculate perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids.

• Circle facts: $C=2\pi r=\pi d$ and $A=\pi r^2$, with the radius half the diameter; using $\pi r$ or $\pi r^2$ for a circumference scores nothing. A composite area is a sum of rectangles, triangles, trapezia and parts of circles, with a cut-out subtracted; show each area separately because each earns a process mark, and keep unrounded values until the end.
• Perimeter counts only the outside edge, so a semicircular end contributes $\pi r$ plus nothing for the diameter joined to the rest.
• Volumes: cone or pyramid $\frac13\times\text{base area}\times\text{height}$, sphere $\frac43\pi r^3$, with the formulae given on the paper.
• Surface area of a cone is the curved part $\pi rl$, $l$ the slant height, plus the base $\pi r^2$; a hemisphere's is $2\pi r^2$ plus its circular base.
• Exact answers stay as multiples of $\pi$.

What the examiner wants. The right formula at the first step. The June 2024 Paper 1H report says many with some knowledge of sectors went wrong immediately by using $\pi r^2$ or $\pi r$ instead of $2\pi r$ and gained no marks, while those who made a correct start usually finished; the Paper 3H report records volume-of-cylinder attempts that found the surface area or multiplied radius by height instead.

Exam tip. Underline 'curved' or 'total' surface area, then list the exposed faces before calculating.

G18 · Calculate arc lengths, angles and areas of sectors of circles

• A sector or arc occupies the same fraction of a circle as its central angle occupies of $360^\circ$. For angle $\theta$, arc length is $\frac{\theta}{360}\times2\pi r$ and sector area is $\frac{\theta}{360}\times\pi r^2$.
• Rearrange these relationships when the angle or radius is unknown.
• An arc length contains only the curved boundary; a sector perimeter also contains the two radii.
• For an annular sector, subtract the inner sector area from the outer one and include both arcs in the perimeter.
• Examiners expect exact answers in terms of $\pi$ unless a decimal accuracy is specified.

Exam tip. Write the fraction $\theta/360$ first; it earns the method whether the question asks for an arc, area or rearranged angle.

G19 · Apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures

• Congruent figures have the same size and shape. Similar figures have equal corresponding angles and corresponding lengths in one constant ratio.
• Match sides by their positions opposite equal angles, then calculate a scale factor as image length divided by original length.
• Multiply every original length by the same factor, or divide to reverse the enlargement.
• Higher tier: if the length scale factor is $k$, the area scale factor is $k^2$ and the volume scale factor is $k^3$.
• Examiners expect corresponding quantities to be paired consistently; a ratio written in the wrong direction gives every later value incorrectly.

Exam tip. Write the scale-factor direction in words, such as 'small to large', before multiplying or dividing.

G20 · Know Pythagoras' theorem a² + b² = c² and the trigonometric ratios sin, cos and tan; apply them to find angles and lengths in right-angled and, where possible, general triangles in 2D and 3D figures

Definition. Specification G20: know the formulae for Pythagoras' theorem $a^2+b^2=c^2$ and the trigonometric ratios $\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}$ and $\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}$; apply them to find angles and lengths in right-angled triangles and, where possible, general triangles in two- and three-dimensional figures.

• Label the triangle first: the hypotenuse is opposite the right angle, and opposite and adjacent are named from the angle in use. Pythagoras finds a side from two others: add the squares for the hypotenuse, subtract for a shorter side, so $x^2=8.5^2-4^2$ gives $x=7.5$.
• A trigonometric ratio links one angle and two sides; write the statement with the values in it, such as $\cos53^\circ=\dfrac{x}{14.5}$, then rearrange, and use the inverse function for an angle. Round only at the end and give at least three significant figures unless told otherwise.
• To test whether a triangle is right-angled, compare the sum of the two smaller squares with the largest square and state the conclusion.
• Higher tier: in three dimensions find a right-angled triangle inside the solid, often needing Pythagoras twice; the diagonal of a cuboid is $\sqrt{a^2+b^2+c^2}$.
• Non-right-angled triangles use the sine and cosine rules.

What the examiner wants. The right version of the formula for the side being found. The June 2024 Paper 2H report calls a plain Pythagoras question a positive start for most, but notes a proportion who mixed up the formula and worked as if finding the hypotenuse; the November 2024 Paper 2H report adds that many forgot to halve the base when using Pythagoras for the height of an isosceles triangle.

Exam tip. For a multi-step triangle question, keep unrounded calculator values for later steps and round only the final answers.

G21 · Know the exact values of sin θ and cos θ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tan θ for θ = 0°, 30°, 45° and 60°

• Know exact sine and cosine values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$, and exact tangent values through $60^\circ$. The sequences are $\sin\theta:0,\frac12,\frac{\sqrt2}{2},\frac{\sqrt3}{2},1$ and $\cos\theta:1,\frac{\sqrt3}{2},\frac{\sqrt2}{2},\frac12,0$.
• Also $\tan0^\circ=0$, $\tan30^\circ=\frac1{\sqrt3}$, $\tan45^\circ=1$ and $\tan60^\circ=\sqrt3$. The $1:\sqrt3:2$ and $1:1:\sqrt2$ triangles explain these values by pairing opposite, adjacent and hypotenuse sides with the marked angle.
• If a value is forgotten, reconstruct it from the appropriate special triangle rather than using a decimal.
• Use exact values throughout multi-step area or algebra calculations.
• Examiners require fractions and surds.

Exam tip. Write each exact trig value before substitution; the unsimplified exact form usually secures the method.

G22 · Know and apply the sine rule a/sin A = b/sin B = c/sin C, and cosine rule a² = b² + c² - 2bc cos A, to find unknown lengths and angles [Higher only]

• Higher tier only. The sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ pairs each side with its opposite angle.
• Use it when a complete opposite side-angle pair is known. The cosine rule $a^2=b^2+c^2-2bc\cos A$ finds a side from two sides and their included angle, or an angle from three sides.
• Label the triangle before substituting so the lowercase side is opposite its matching capital angle.
• When an inverse sine gives an angle, check the supplementary value against the triangle angle sum and other information.
• Examiners expect an unrounded substitution followed by the requested final accuracy.

Exam tip. Sketch and label $a$ opposite $A$ before choosing a rule; this prevents most substitution errors.

G23 · Know and apply Area = ½ ab sin C to calculate the area, sides or angles of any triangle [Higher only]

• Higher tier only. Use $A=\frac12ab\sin C$ when two sides $a$ and $b$ and their included angle $C$ are known.
• The formula comes from $\frac12\times\text{base}\times\text{perpendicular height}$, with height $b\sin C$.
• Rearrange it to find a missing side or use inverse sine for an angle.
• Because $\sin C=\sin(180^\circ-C)$, an angle calculation may have both an acute and an obtuse solution; test both against the stated triangle.
• Examiners expect the angle between the two substituted sides, correct square units for area, and all valid angle solutions unless the diagram or context excludes one.

Exam tip. On the diagram, circle the included angle between the two substituted sides before using $\frac12ab\sin C$.

G24 · Describe translations as 2D vectors

• A translation moves every point by the same horizontal and vertical displacement, preserving lengths, angles and orientation. Describe it with a column vector $\binom{x}{y}$: a positive top component moves right and a negative one moves left; a positive bottom component moves up and a negative one moves down.
• To recover the vector from object and image coordinates, calculate image minus object in each coordinate.
• Apply the same vector to every vertex when constructing an image.
• Examiners expect a vector, not a destination coordinate, and the object-to-image direction matters.
• Matching one pair of corresponding points is enough if the shapes are genuinely translations.

Exam tip. Use image minus object for both coordinates, then check the signs against the visible direction of movement.

G25 · Apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; use vectors to construct geometric arguments and proofs

• A vector has magnitude and direction and may be shown as a directed segment, a column vector or a symbol such as $\mathbf a$. Add column vectors component by component.
• Subtracting a vector adds its reverse, while multiplying by scalar $k$ multiplies every component and reverses direction when $k<0$.
• For a route through successive points, add vectors in travel order; position-vector differences use destination minus start.
• Higher tier: compare alternative routes, or show one direction vector is a scalar multiple of another, to prove points are collinear or lines parallel.
• Examiners expect a connected vector equation and an explicit geometric conclusion.

Exam tip. In a vector proof, finish with words such as 'therefore parallel' or 'therefore collinear' after the scalar-multiple equation.