Eduqas GCSE Mathematics Geometry and measures: a complete overview of angles, trigonometry, area and volume, circle theorems, transformations and vectors

What the Geometry and measures content demands

Geometry and measures is the largest content area, mixing precise drawing, formula recall and sustained reasoning. Eduqas tests both calculation and justification here, especially in angle and circle-theorem questions where the named reason carries marks, so a fluent geometric vocabulary matters as much as the arithmetic. The content runs from angle facts through trigonometry and measurement to circle theorems, transformations and vectors, with the most demanding topics (the sine and cosine rules, circle theorems, vector proofs) reserved for Higher tier. Some formulae are given and some must be recalled.

This guide walks through the seven areas of the content and ties together the matching dot-point pages, each with its own practice questions.

Angles and polygons

Angles on a line sum to $180^\circ$, around a point to $360^\circ$, and vertically opposite angles are equal. Across parallel lines, corresponding (F) and alternate (Z) angles are equal while co-interior (C) angles sum to $180^\circ$. A polygon's exterior angles sum to $360^\circ$ and its interior angles to $(n - 2) \times 180^\circ$. Eduqas's "give reasons" questions reward naming the fact used at each step.

Pythagoras and trigonometry

Pythagoras $a^2 + b^2 = c^2$ finds a side of a right-angled triangle. SOH CAH TOA links angles to side ratios for finding sides and angles. At Higher tier, the sine rule (a side-angle pair), the cosine rule (two sides and the included angle, or three sides) and the area formula $\tfrac{1}{2}ab\sin C$ extend trigonometry to any triangle; all three are on the formulae list.

Area and volume

Recall the area formulae (rectangle $bh$, triangle $\tfrac{1}{2}bh$, trapezium $\tfrac{1}{2}(a+b)h$, circle $\pi r^2$) and the circumference $2\pi r$. A sector is a fraction $\tfrac{\theta}{360}$ of the circle. Volume of a prism is cross-section times length; the cylinder, sphere and cone follow their formulae (sphere and cone given). Surface area adds every face. Units are squared for area and cubed for volume.

Circles and circle theorems

At Higher tier, the circle theorems relate angles formed by chords, radii, tangents and arcs: 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. Every step in a proof needs its theorem named.

Transformations

Translation slides by a vector; rotation turns by an angle, direction and centre; reflection flips in a mirror line; enlargement scales by a factor from a centre. Negative scale factors (Higher) invert the image through the centre, and fractional factors shrink it. A full description must give every defining detail of the transformation.

Vectors

A vector has magnitude and direction, written as a column vector or $\overrightarrow{AB}$. Add and subtract component by component, and scale by multiplying each component. Vector geometry finds a path across a diagram in terms of named vectors, and at Higher tier proves lines parallel (one vector a scalar multiple of another) or points collinear.

Constructions and loci

The ruler-and-compass constructions (perpendicular bisector, angle bisector, perpendicular from a point) must be drawn accurately with the arcs left showing. A locus is the set of points satisfying a condition: a circle (fixed distance from a point), parallel lines (fixed distance from a line), a perpendicular bisector (equidistant from two points) or an angle bisector (equidistant from two lines). Combined conditions give a region.

Check your knowledge

A mix of angle, trigonometry, measurement and transformation questions. Attempt them under timed conditions, then check against the solutions.

1. A regular octagon. Find the size of each interior angle. (2 marks)
2. A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse. (2 marks)
3. Find the area of a circle of radius 6 cm, in terms of $\pi$. (1 mark)
4. Find the volume of a cylinder of radius 4 cm and height 10 cm, in terms of $\pi$. (2 marks)
5. The angle at the centre of a circle is $80^\circ$. Find the angle at the circumference on the same arc. (1 mark)
6. Describe the transformation given by the column vector $\begin{pmatrix} -3 \\ 5 \end{pmatrix}$. (1 mark)
7. $\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$. Find $\mathbf{a} + 2\mathbf{b}$. (2 marks)
8. What is the locus of points exactly 5 cm from a fixed point P? (1 mark)