2D Shapes
2D shapes are flat, closed figures with length and width but no depth. Understanding their properties is essential for solving problems involving area, perimeter, angles, and transformations. This guide covers key shapes, their properties, and important formulae you’ll need for the GCSE Maths exam.
Types of 2D Shapes and Their Properties
Polygons
A polygon is a 2D shape with straight sides. The number of sides determines its name and properties.
| Shape | Number of Sides | Key Properties |
|---|---|---|
| Triangle | 3 | Interior angles sum to 180° |
| Quadrilateral | 4 | Interior angles sum to 360° |
| Pentagon | 5 | Interior angles sum to 540° |
| Hexagon | 6 | Interior angles sum to 720° |
| Heptagon | 7 | Interior angles sum to 900° |
| Octagon | 8 | Interior angles sum to 1,080° |
For any n-sided polygon, the sum of the interior angles is:
$$\text{Sum of interior angles} = (n - 2) \times 180^\circ$$
The exterior angles of any polygon always add up to 360°.
Triangles
Triangles are classified by their angles and side lengths:
| Type | Properties |
|---|---|
| Equilateral | Three equal sides, three equal angles (60° each) |
| Isosceles | Two equal sides, two equal angles |
| Scalene | No equal sides, no equal angles |
| Right-angled | One angle is 90° |
Pythagoras’ Theorem (for right-angled triangles):
$$a^2 + b^2 = c^2$$
(where $c$ is the hypotenuse, the longest side).
Trigonometry (for right-angled triangles):
$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$
Quadrilaterals
There are several types of quadrilaterals with unique properties:
| Shape | Properties |
|---|---|
| Square | Four equal sides, four right angles (90°) |
| Rectangle | Opposite sides equal, four right angles |
| Rhombus | Four equal sides, opposite angles equal |
| Parallelogram | Opposite sides equal and parallel, opposite angles equal |
| Trapezium | One pair of parallel sides |
| Kite | Two pairs of adjacent equal sides, one pair of equal angles |
This video explains the properties of Triangles and Quadrilaterals
Area and Perimeter Formulae
Basic Shapes
| Shape | Area Formula | Perimeter Formula |
|---|---|---|
| Square | $A = s^2$ | $P = 4s$ |
| Rectangle | $A = l \times w$ | $P = 2(l + w)$ |
| Triangle | $A = \frac{1}{2} b h$ | Sum of all sides |
| Trapezium | $A = \frac{1}{2} (a + b) h$ | Sum of all sides |
| Parallelogram | $A = b \times h$ | Sum of all sides |
| Rhombus | $A = \frac{1}{2} d_1 d_2$ | Sum of all sides |
Circles
Circles are not polygons but are important 2D shapes.
| Formula | Expression |
|---|---|
| Circumference | $C=2πr$ or $C=πd$ |
| Area | $A = \pi r^2$ |
Where:
- $r$ = radius
- $d$ = diameter ($d=2r$)
- $\pi \approx 3.1416$
Angle Rules in 2D Shapes
Triangle Angle Rules
- Angles in a triangle add up to 180°.
- Exterior angle = sum of the two opposite interior angles.
- Base angles of an isosceles triangle are equal.
Quadrilateral Angle Rule
- Angles in a quadrilateral add up to 360°.
Parallel Line Rules
These are useful for working with polygons and transformations.
| Rule | Description |
|---|---|
| Corresponding angles | Equal ($F$-shape) |
| Alternate angles | Equal ($Z$-shape) |
| Co-interior angles | Add up to 180° ($C$-shape) |
Transformations
2D shapes can be transformed in four main ways:
| Transformation | Description |
|---|---|
| Translation | Moves a shape without rotating or flipping it (described by a vector $\begin{pmatrix} x \\ y \end{pmatrix}$) |
| Reflection | Flips a shape across a mirror line (e.g., $x$-axis, $y$-axis). |
| Rotation | Turns a shape around a fixed point (clockwise or anticlockwise by $90°,180°,270°$). |
| Enlargement | Changes the size of a shape by a scale factor from a centre of enlargement. |
If the scale factor is:
- Greater than 1, the shape enlarges.
- Between 0 and 1, the shape shrinks.
- Negative, the shape is also rotated by 180°.
Exam Tips
✔ Memorise key formulae, but also understand how to apply them.
✔ Practise problem-solving—many questions involve multiple concepts at once.
✔ Use a ruler and protractor for accurate diagrams.
✔ Check your working—especially when using angle rules or transformations.
✔ Show full working to get method marks, even if the final answer is incorrect.
Summary Table of Key Formulae
| Topic | Formula |
|---|---|
| Interior angles sum of polygon | $(n - 2) \times 180^\circ$ |
| Exterior angles sum of polygon | $360°$ |
| Pythagoras’ Theorem | $a^2 + b^2 = c^2$ |
| Trigonometry (SOHCAHTOA) | $\sin \theta = \frac{o}{h}, \cos \theta = \frac{a}{h}, \tan \theta = \frac{o}{a}$ |
| Area of circle | $\pi r^2$ |
| Circumference of circle | $\pi r2πr$ or $πd$ |
This guide covers the essential topics for 2D shapes in GCSE Maths. Keep practising, and you’ll be well prepared for your exams.
