OCR GCSE Mathematics Geometry and measures: a complete overview of angles, Pythagoras, trigonometry, mensuration, circles, transformations and vectors

What the Geometry content demands

Geometry and measures is the largest content area of OCR GCSE Mathematics, and it rewards both accurate calculation and clear geometric reasoning. The content runs from angle facts through Pythagoras, trigonometry and mensuration to circle theorems, transformations, vectors and constructions. Because so many questions ask you to "give a reason" or "describe fully", this area carries a large share of the AO2 reasoning marks.

This guide walks through the seven areas of the Geometry content 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. Across parallel lines, alternate and corresponding angles are equal and co-interior angles sum to $180^\circ$. A polygon's exterior angles sum to $360^\circ$ and its interior angles to $(n - 2) \times 180^\circ$. Each step in an angle chase needs its named reason, which OCR marks directly.

Pythagoras and trigonometry

Pythagoras' theorem $a^2 + b^2 = c^2$ finds a side of a right-angled triangle. The ratios are $\sin\theta = \tfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \tfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \tfrac{\text{opp}}{\text{adj}}$ (SOHCAHTOA), with inverse functions for angles. For any triangle at Higher tier, use the sine rule, the cosine rule and the area formula $\tfrac{1}{2}ab\sin C$, all on the formulae sheet.

Area and volume

Area of a rectangle is $lw$, a triangle $\tfrac{1}{2}bh$, a trapezium $\tfrac{1}{2}(a + b)h$; a circle has area $\pi r^2$ and circumference $2\pi r$, and a sector is a fraction of the circle. Volume of a prism is cross-section times length, a cylinder is $\pi r^2 h$, and the sphere $\tfrac{4}{3}\pi r^3$ and cone $\tfrac{1}{3}\pi r^2 h$ are given on the formulae sheet. Watch units: area squared, volume cubed.

Circles and circle theorems

At Higher tier, the circle theorems relate angles, chords and tangents: the angle at the centre is twice that at the circumference, the angle in a semicircle is $90^\circ$, angles in the same segment are equal, cyclic-quadrilateral opposite angles sum to $180^\circ$, a tangent meets a radius at $90^\circ$, and the alternate segment theorem. Each must be named when used.

Transformations

A translation uses a column vector; a rotation needs angle, direction and centre; a reflection needs the mirror-line equation; an enlargement needs a scale factor and centre, with fractional factors shrinking and negative factors inverting. OCR marks the completeness of the description, and area scales by the square of any length scale factor.

Vectors

Add and subtract vectors component by component, and scale by multiplying each component. In vector geometry, build a path nose to tail, so $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$. Parallel vectors are scalar multiples, which is the key to proving lines parallel or points collinear, a demanding Higher-tier skill.

Constructions and loci

The standard constructions (perpendicular bisector, angle bisector, perpendicular from a point) use ruler and compasses with the arcs left visible, because they carry marks. A locus is the set of points meeting a condition: a fixed distance from a point is a circle, from a line a pair of parallels, and equidistant from two points the perpendicular bisector. Combine loci to shade a region.

Check your knowledge

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

1. Find the size of each interior angle of a regular octagon. (3 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 $5$ cm, in terms of $\pi$. (1 mark)
4. The angle at the centre of a circle is $86^\circ$. Find the angle at the circumference on the same arc. (1 mark)
5. Work out the volume of a cylinder with radius $3$ cm and height $10$ cm, in terms of $\pi$. (2 marks)
6. A triangle is enlarged by scale factor $4$. By what factor does its area increase? (1 mark)
7. Vectors $\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$. Find $\mathbf{a} - \mathbf{b}$. (2 marks)
8. Find the exterior angle of a regular pentagon. (1 mark)