WJEC GCSE Mathematics Geometry and measures: a complete overview of angles, mensuration, circles, trigonometry, transformations, constructions, similarity and vectors

What the Geometry and measures content demands

Geometry and measures is the largest content area after Number and Algebra, and it rewards accurate, labelled diagrams and clearly reasoned working. WJEC examines geometric reasoning heavily under AO2 and AO3, so "find the angle, giving a reason" and "prove that" questions carry marks for the justification, not just the value. Because Unit 1 is non-calculator, exact and mental geometry matters, while Unit 2 brings in trigonometry and mensuration that need a calculator. This guide walks through the content and links to the matching dot-point pages, each with its own practice questions.

Angles and polygons

The basic angle facts (angles at a point sum to $360^\circ$, on a line to $180^\circ$, vertically opposite angles equal), the angle sums of triangles ($180^\circ$) and quadrilaterals ($360^\circ$), and the parallel line rules (corresponding and alternate angles equal, co-interior angles supplementary) solve most angle problems. Polygons add the interior angle sum $(n-2)\times 180^\circ$ and the constant exterior angle sum of $360^\circ$. Always state the reason for each step.

Area and volume

Know the area formulae for rectangles, triangles, parallelograms and trapezia, the circle formulae $C = 2\pi r$ and $A = \pi r^2$, and sector arc length and area as fractions $\tfrac{\theta}{360^\circ}$ of the circle. For solids, a prism is cross-section area times length, a cylinder is $\pi r^2 h$, a sphere is $\tfrac{4}{3}\pi r^3$ and a cone is $\tfrac{1}{3}\pi r^2 h$. Compound shapes split into known parts, and units must be square for area and cubic for volume.

Circles and circle theorems (Higher)

Beyond the parts of a circle, the Higher circle theorems relate angles: the angle at the centre is twice the angle at the circumference, the angle in a semicircle is $90^\circ$, angles in the same segment are equal, opposite angles of a cyclic quadrilateral sum to $180^\circ$, a tangent meets a radius at $90^\circ$, two tangents from a point are equal, and the alternate segment theorem. Name the theorem at every step.

Pythagoras and trigonometry

Pythagoras' theorem $a^2 + b^2 = c^2$ finds a side of a right-angled triangle, and SOH CAH TOA finds sides and angles. At Higher tier the sine rule $\tfrac{a}{\sin A} = \tfrac{b}{\sin B} = \tfrac{c}{\sin C}$, the cosine rule $a^2 = b^2 + c^2 - 2bc\cos A$ and the area formula $\tfrac{1}{2}ab\sin C$ extend trigonometry to any triangle. Choosing the right tool from the given information is the marked judgement.

Transformations

Translation slides by a column vector, reflection flips in a stated line, rotation turns by an angle and direction about a centre, and enlargement scales by a scale factor from a centre (negative and fractional at Higher). The recurring command "describe fully the single transformation" needs every defining detail, and a combination of transformations can often be replaced by one equivalent transformation.

Constructions, loci, similarity and vectors

Ruler-and-compass constructions (perpendicular and angle bisectors, the perpendicular from a point) double as the basic loci, and visible arcs earn the marks. Bearings are three-figure angles clockwise from north. Similar shapes share angles with sides in ratio $k$, with areas scaling by $k^2$ and volumes by $k^3$; congruent shapes match by SSS, SAS, ASA or RHS. Vectors (Higher) use column notation, add and scale component by component, and prove lines parallel or points collinear when one vector is a scalar multiple of another.

Check your knowledge

A mix of angle, mensuration, trigonometry, transformation and similarity questions covering the Geometry and measures content. Attempt them under timed conditions, then check against the solutions.

1. The exterior angle of a regular polygon is $24^\circ$. How many sides has it? (2 marks)
2. Find the area of a circle of radius $7$ cm, in terms of $\pi$. (2 marks)
3. A right-angled triangle has shorter sides $9$ cm and $12$ cm. Find the hypotenuse. (2 marks)
4. Find the volume of a cylinder with radius $3$ cm and height $10$ cm, in terms of $\pi$. (2 marks)
5. Describe fully the transformation that maps a point $(x, y)$ to $(-x, -y)$. (2 marks)
6. Two similar shapes have linear scale factor $3$. By what factor do their areas differ? (1 mark)
7. Given $\mathbf{a} = \binom{2}{5}$ and $\mathbf{b} = \binom{1}{-3}$, find $\mathbf{a} + 2\mathbf{b}$. (2 marks)
8. State the circle theorem that gives the angle in a semicircle. (1 mark)