Edexcel GCSE Mathematics Geometry and measures: a complete overview of angles, trigonometry, area, circles, transformations and vectors

What this content demands

Geometry and measures is the most visual area of the course, and one of the largest. Edexcel tests precise angle reasoning with named reasons, fluent use of Pythagoras and trigonometry, accurate area and volume work, the circle theorems, transformations, vectors and constructions. Many questions ask you to justify each step, so clear method matters as much as the final number.

This guide walks through the seven areas and ties together the matching dot-point pages, each of which has 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. In parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles add to $180^\circ$. A polygon's interior angles sum to $(n - 2) \times 180^\circ$, and its exterior angles always sum to $360^\circ$. On "give reasons" questions, name every rule.

Pythagoras and trigonometry

$a^2 + b^2 = c^2$ links the sides of a right-angled triangle. SOH CAH TOA gives sides and angles from the ratios. Know the exact values of sin, cos and tan for $30^\circ$, $45^\circ$ and $60^\circ$. For any triangle (Higher), use the sine rule with an opposite pair, the cosine rule with two sides and the included angle, and the area formula $\tfrac{1}{2}ab\sin C$.

Area and volume

Triangle $\tfrac{1}{2}bh$, parallelogram $bh$, trapezium $\tfrac{1}{2}(a + b)h$, circle area $\pi r^2$ and circumference $2\pi r$. The volume of any prism is cross-section $\times$ length. Higher tier adds cone $\tfrac{1}{3}\pi r^2 h$, sphere $\tfrac{4}{3}\pi r^3$ and pyramid $\tfrac{1}{3} \times \text{base} \times \text{height}$, plus surface areas.

Circles and circle theorems

An arc and sector are fractions $\dfrac{\theta}{360}$ of the circumference and area. The circle theorems (Higher) include the angle at the centre being twice that at the circumference, the angle in a semicircle being $90^\circ$, equal angles in the same segment, cyclic quadrilaterals summing to $180^\circ$, the tangent-radius right angle, and the alternate segment theorem.

Transformations

Translation (by a vector), reflection (in a named line), rotation (angle, direction, centre) and enlargement (scale factor, centre). Fractional factors shrink and negative factors flip to the opposite side. Areas scale by the square of the factor. Describe a single transformation with all its required details.

Vectors

Column vectors add, subtract and scale component by component, with magnitude $\sqrt{x^2 + y^2}$. In geometry, $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, and proofs show one vector is a scalar multiple of another to establish parallel lines or collinear points.

Constructions and loci

The perpendicular bisector, angle bisector and perpendicular from a point are built with ruler and compasses, leaving the arcs visible. Loci are sets of points: 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), often combined to shade a region.

Check your knowledge

A mix of angle, measure and reasoning questions. Attempt them under timed conditions, then check against the solutions.

1. A regular polygon has an exterior angle of $30^\circ$. How many sides does it have? (2 marks)
2. Find the hypotenuse of a right-angled triangle with legs $9\,\text{cm}$ and $12\,\text{cm}$. (2 marks)
3. Work out the area of a circle with radius $7\,\text{cm}$ in terms of $\pi$. (1 mark)
4. A sector has radius $10\,\text{cm}$ and angle $72^\circ$. Find its area in terms of $\pi$. (2 marks)
5. The angle at the centre of a circle is $140^\circ$. Find the angle at the circumference on the same arc. (1 mark)
6. Describe fully the transformation given by the vector $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$. (1 mark)
7. Find the magnitude of the vector $\begin{pmatrix} 8 \\ 6 \end{pmatrix}$. (2 marks)
8. Describe the locus of points equidistant from two fixed points $A$ and $B$. (1 mark)