Geometry is the mathematics of shape, size, position and space. It is the largest topic on the GCSE paper, but it rests on a surprisingly small number of core ideas.
• Angle facts — a handful of rules about angles that combine to solve almost any angle problem.
• Properties of shapes — knowing what makes a rhombus a rhombus lets you deduce lengths and angles for free.
• Formulae — for length, area and volume, which you apply and combine.
• Right-angled triangles — Pythagoras and trigonometry, which turn a picture into an equation.
| Fact | Reason to write |
|---|---|
| Angles on a straight line add to $180^\circ$ | angles on a straight line |
| Angles round a point add to $360^\circ$ | angles at a point |
| Vertically opposite angles are equal | vertically opposite angles |
| Angles in a triangle add to $180^\circ$ | angle sum of a triangle |
| Angles in a quadrilateral add to $360^\circ$ | angle sum of a quadrilateral |
| Alternate angles are equal (Z-shape) | alternate angles |
| Corresponding angles are equal (F-shape) | corresponding angles |
| Co-interior angles add to $180^\circ$ (C-shape) | co-interior angles |
| Base angles of an isosceles triangle are equal | base angles of an isosceles triangle |
| Exterior angle of a triangle $=$ sum of the two opposite interior angles | exterior angle of a triangle |
Most of these are not given in the exam, so they must be learnt.
| Shape | Formula |
|---|---|
| Triangle | Area $= \tfrac{1}{2}bh$ |
| Parallelogram | Area $= bh$ |
| Trapezium | Area $= \tfrac{1}{2}(a+b)h$ |
| Circle | $C = 2\pi r = \pi d$ · $A = \pi r^2$ |
| Any prism | Volume $=$ area of cross-section $\times$ length |
| Cylinder | $V = \pi r^2 h$ · curved surface $= 2\pi r h$ |
| Cone | $V = \tfrac{1}{3}\pi r^2 h$ · curved surface $= \pi r l$ |
| Sphere | $V = \tfrac{4}{3}\pi r^3$ · surface $= 4\pi r^2$ |
| Pyramid | $V = \tfrac{1}{3} \times$ base area $\times$ height |
| Right-angled triangle | $a^2 + b^2 = c^2$ · SOH CAH TOA |
| Any triangle (Higher) | $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$ · $a^2 = b^2 + c^2 - 2bc\cos A$ · Area $= \tfrac{1}{2}ab\sin C$ |
The National Curriculum divides Geometry and Measures into 25 statements, usually labelled G1 to G25 in three groups. Each has its own page with explanations, worked examples and ten practice questions.
Properties and constructions (G1–G13)
- G1Geometric Conventions and NotationPoints, lines, planes, labelling triangles, symmetry, drawing from a description.
- G2Ruler and Compass Constructions and LociBisectors, perpendiculars, and regions satisfying a set of conditions.
- G3Angle Rules and PolygonsAngles at a point and on lines, parallel-line angles, interior and exterior angles.
- G4Properties of Triangles and QuadrilateralsDefinitions and properties of every named triangle and quadrilateral.
- G5Congruent TrianglesThe four criteria SSS, SAS, ASA and RHS, and how to write a congruence proof.
- G6Geometric Reasoning and ProofChaining angle facts and congruence into a full written proof.
- G7TransformationsRotation, reflection, translation and enlargement, including negative scale factors.
- G8Combined Transformations and InvarianceDescribing a single transformation equivalent to two, and what stays fixed.
- G9Parts of a CircleRadius, chord, tangent, arc, sector, segment and their basic properties.
- G10Circle TheoremsAll eight theorems, their proofs, and how to apply them. (Higher)
- G11Coordinate GeometryMidpoints, distances, gradients, parallel and perpendicular lines, shapes on axes.
- G12Properties of 3D ShapesFaces, edges and vertices of prisms, pyramids, cylinders, cones and spheres.
- G13Plans and ElevationsDrawing and interpreting 2D views of a 3D solid.
Mensuration and calculation (G14–G22)
- G14Standard Units of MeasureLength, area, volume, mass, time and money, and converting between them.
- G15Measuring, Scale Drawings and BearingsUsing a ruler and protractor, maps, and three-figure bearings.
- G16Area and Volume FormulaeTriangles, parallelograms, trapezia, cuboids and other prisms.
- G17Circles, Surface Area and VolumeCircumference and area of circles; spheres, cones, pyramids and composite solids.
- G18Arcs and SectorsArc length, sector area, segment area and perimeters of sectors.
- G19Congruence and Similarity in MeasuresLength, area and volume scale factors $k$, $k^2$ and $k^3$.
- G20Pythagoras and TrigonometryRight-angled triangles in 2D and 3D, SOH CAH TOA, and finding angles.
- G21Exact Trigonometric ValuesThe exact values for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$.
- G22Sine Rule, Cosine Rule and Area of a TriangleSolving any triangle, not just right-angled ones. (Higher)
Vectors (G23–G25)
- G23Vectors and TranslationsColumn vectors, notation, magnitude and describing translations.
- G24Vector ArithmeticAdding, subtracting, multiplying by a scalar, and vector diagrams.
- G25Vector Geometry and ProofUsing vectors to prove that lines are parallel or that points are collinear. (Higher)
- Mark up the diagram. Write every angle and length you work out straight onto the picture. Use tick marks for equal sides and arcs for equal angles.
- Look for the standard shapes. Isosceles triangles, parallel lines, circles with a radius drawn — each brings a rule with it.
- Work out anything you can, even if it does not look useful yet. Geometry questions unlock a step at a time.
- Write the reason for every step. Marks are given for reasons, not just numbers.
- Check the answer is sensible. An angle that looks acute should not come out as $130^\circ$.
In triangle $ABC$, $AB = AC$ and angle $BAC = 40^\circ$. The line $BC$ is extended to $D$. Find angle $ACD$.
Answer: $110^\circ$ — and note the reason given at every stage.
Alternatively: the exterior angle of a triangle equals the sum of the two opposite interior angles, so angle $ACD = 40 + 70 = 110^\circ$ ✓
Unless it says "accurately drawn", never measure. Two lines that look equal may not be.
Exterior angles always add to $360^\circ$; interior angles add to $(n-2)\times 180^\circ$.
Area of a triangle or parallelogram needs the perpendicular height, not a sloping side.
Lengths $\times k$, areas $\times k^2$, volumes $\times k^3$.
SOH CAH TOA only works when there is a right angle. Otherwise you need the sine or cosine rule.
Keep full accuracy on your calculator throughout, and round only at the very end.
Angle sums
Straight line $180^\circ$, point $360^\circ$, triangle $180^\circ$, quadrilateral $360^\circ$.
Polygons
Interior sum $= (n-2)\times 180^\circ$; exterior sum $= 360^\circ$.
Parallel lines
Alternate equal, corresponding equal, co-interior add to $180^\circ$.
Congruence
SSS, SAS, ASA, RHS. Never SSA.
Pythagoras
$a^2 + b^2 = c^2$, with $c$ the hypotenuse.
Trigonometry
$\sin = \tfrac{O}{H}$, $\cos = \tfrac{A}{H}$, $\tan = \tfrac{O}{A}$.
Circle
$C = 2\pi r$, $A = \pi r^2$. Arc and sector are fractions of these.
Similar shapes
Length $k$, area $k^2$, volume $k^3$.
Vectors
$\begin{pmatrix} x \\ y \end{pmatrix}$: $x$ across, $y$ up. Add by adding components.
Always
Give a reason for every step of an angle chase.
These ten questions sample the whole topic. If one type catches you out, follow the link in Section 4 to the page that covers it.
Three angles on a straight line are $x$, $2x$ and $3x$. Find $x$.
▶ Show solution
Angles on a straight line add to $180^\circ$.
$x + 2x + 3x = 180 \Rightarrow 6x = 180$
$x = 30^\circ$
Find the size of each interior angle of a regular decagon (10 sides).
▶ Show solution
Exterior angle $= 360 \div 10 = 36^\circ$.
Interior angle $= 180 - 36 = 144^\circ$.
Check: interior sum $= (10-2)\times 180 = 1440^\circ$; $1440 \div 10 = 144^\circ$ ✓
A right-angled triangle has legs $9$ cm and $12$ cm. Find the hypotenuse.
▶ Show solution
$c^2 = 9^2 + 12^2 = 81 + 144 = 225$
$c = \sqrt{225} = 15$ cm
A circle has radius $7$ cm. Find its circumference and area, each to 1 decimal place.
▶ Show solution
$C = 2\pi r = 2 \times \pi \times 7 = 43.98\ldots = 44.0$ cm
$A = \pi r^2 = \pi \times 49 = 153.93\ldots = 153.9\text{ cm}^2$
Find the area of a trapezium with parallel sides $8$ cm and $14$ cm and perpendicular height $5$ cm.
▶ Show solution
$A = \tfrac{1}{2}(a+b)h = \tfrac{1}{2}(8+14)\times 5$
$= \tfrac{1}{2} \times 22 \times 5 = 55\text{ cm}^2$
In a right-angled triangle the hypotenuse is $10$ cm and one angle is $35^\circ$. Find the side opposite that angle, to 1 d.p.
▶ Show solution
Opposite and hypotenuse → use sine.
$\sin 35^\circ = \dfrac{O}{10}$
$O = 10 \sin 35^\circ = 10 \times 0.57357 = 5.7$ cm
A cylinder has radius $4$ cm and height $10$ cm. Find its volume to 1 d.p.
▶ Show solution
$V = \pi r^2 h = \pi \times 16 \times 10 = 160\pi$
$= 502.65\ldots = 502.7\text{ cm}^3$
Describe fully the single transformation that maps the point $(3,\ 2)$ to $(3,\ -2)$, and maps $(5,\ 1)$ to $(5,\ -1)$.
▶ Show solution
The $x$-coordinates are unchanged and the $y$-coordinates have changed sign.
That is a reflection in the $x$-axis (the line $y = 0$).
$\mathbf{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}$. Find $2\mathbf{a} + \mathbf{b}$.
▶ Show solution
$2\mathbf{a} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}$
$2\mathbf{a} + \mathbf{b} = \begin{pmatrix} 6 + (-2) \\ -2 + 5 \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$
Two similar cones have heights $6$ cm and $9$ cm. The smaller has volume $96\text{ cm}^3$ and surface area $80\text{ cm}^2$.
(a) Find the volume of the larger. (b) Find its surface area.
▶ Show solution
Length scale factor $k = \dfrac{9}{6} = 1.5$.
(a) Volume factor $= 1.5^3 = 3.375$; $96 \times 3.375 = 324\text{ cm}^3$.
(b) Area factor $= 1.5^2 = 2.25$; $80 \times 2.25 = 180\text{ cm}^2$.