Geometry & measures

Angles, polygons, congruence and similarity, Pythagoras, the circle theorems, mensuration and vectors.

11 min read

Geometry rewards knowing your angle facts, area/volume formulae and the circle theorems cold. Most questions are short once you spot which fact applies.

Core facts: angles on a straight line sum to $180°$ , around a point to $360°$ ; vertically opposite angles are equal; with parallel lines, alternate and corresponding angles are equal.

For a polygon with $n$ sides, the interior angles sum to $(n - 2) \times 180°$ and the exterior angles always sum to $360°$ . A regular $n$ -gon has each exterior angle $= \frac{360°}{n}$ .

Worked example

Each interior angle of a regular polygon is $150°$ . How many sides?

Exterior angle $= 180° - 150° = 30°$ , so $n = \frac{360}{30} = 12.$

Pythagoras: in a right-angled triangle $a^{2} + b^{2} = c^{2}$ , with $c$ the hypotenuse. It works in 3D too via a right-angled triangle through the solid.

Similar shapes have equal angles and sides in a fixed ratio (the scale factor $k$ ). Then areas scale by $k^{2}$ and volumes by $k^{3}$ .

Worked example

Two similar cylinders have heights $3 cm$ and $6 cm$ ; the smaller has volume $20 cm^{3}$ . Find the larger volume.

Scale factor $k = 2$ , so volume scales by $k^{3} = 8$ : $20 \times 8 = 160 cm^{3} .$

A circle with centre O and points A, B and P on the circumference; the angle AOB at the centre is twice the angle APB at the circumference. — The angle at the centre is twice the angle at the circumference standing on the same arc.

The standard theorems:

The angle at the centre is twice the angle at the circumference on the same arc.
The angle in a semicircle is $90°$ .
Angles in the same segment are equal.
A tangent meets a radius at $90°$ .
Opposite angles of a cyclic quadrilateral sum to $180°$ .

Worked example

$A B$ is a diameter and $P$ lies on the circle. If angle $P A B = 35°$ , find angle $A B P$ .

The angle in a semicircle gives angle $A P B = 90°$ , so angle $A B P = 180° - 90° - 35° = 55°.$

Know these formulae:

Circle: circumference $= 2 π r = π d$ , area $= π r^{2}$ .
Cylinder: volume $= π r^{2} h$ .
A sector of angle $θ$ : arc $= \frac{θ}{360} \times 2 π r$ , area $= \frac{θ}{360} \times π r^{2}$ .

Formulae for spheres, cones and pyramids are given in the question if needed.

Worked example

A sector has radius $6 cm$ and angle $60°$ . Find its area (leave in terms of $π$ ).

$\frac{60}{360} \times π (6)^{2} = \frac{1}{6} \times 36 π = 6 π cm^{2} .$

A vector has magnitude and direction, written as a column $(x y)$ . Add by adding components; multiply by a scalar to stretch. Parallel vectors are scalar multiples of each other.

Worked example

If $a = (3 - 1)$ and $b = (14)$ , then $2 a - b = (6 - 2) - (14) = (5 - 6) .$

Scaling area or volume by $k$ instead of $k^{2}$ or $k^{3}$ .
Mixing up which circle theorem applies (check what arc the angles stand on).
Using degrees for the sector fraction wrongly: it is always $\frac{θ}{360}$ .

Geometry & measures

Overview

Angles & polygons

Pythagoras, congruence & similarity

Circle theorems

Mensuration

Vectors

Common mistakes