Home

Maths

Shapes and area

Properties of 2D shapes

Select Lesson

Exam Board

Select an option

Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Number

Explainer Video

Tutor: Bilal

Summary

Properties of 2D shapes

In a nutshell

2D shapes such as triangles and quadrilaterals can be further classified into more distinct shapes that each have their own properties. There are also two different types of symmetry that can be applied to 2D shapes - line symmetry and rotational symmetry. Regular polygons can also be classified by their internal and external angles.

Symmetry

There are two different types of symmetry: line symmetry and rotational symmetry.

Line symmetry

Line symmetry of a shape is when you can draw a mirror line across the shape. A mirror line is a line that can be drawn that acts like a mirror; it reflects one side of the line onto the other.

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Rotational symmetry

Rotational symmetry is when you can rotate the shape and it looks exactly the same.

The order of rotational symmetry is a property that tells you how many times the shape looks the same when you rotate it a full circle.

This shape has rotational symmetry of order $2$ because rotating it $180^\circ$ will make it the same shape again.

Regular polygons

A regular polygon is a 2D shape that has the same side lengths and angles.

NUMBER OF SIDES	NAME OF REGULAR POLYGON	SHAPE
$3$	Equilateral triangle
$4$	Square
$5$	Regular pentagon
$6$	Regular hexagon
$7$	Regular heptagon
$8$	Regular octagon
$9$	Regular nonagon
$10$	Regular decagon

Triangles

There are four main types of triangles.

NAME	DESCRIPTION	LINES OF SYMMETRY	ORDER OF ROTATIONAL SYMMETRY	SHAPE
Equilateral triangle	All sides of equal length, all angles are $60^\circ$	$3$	$3$
Right-angled triangle	$1$ angle of $90^\circ$	$0$ , or $1$ if it's a right-angled isoceles triangle	$0$
Isosceles triangle	$2$ equal sides and $2$ equal angles	$1$	$0$
Scalene triangle	No equal sides or angles	$0$	$0$

Quadrilaterals

A quadrilateral is the name given to a shape that has four sides. There are six main types of quadrilaterals.

NAME	DESCRIPTION	LINES OF SYMMETRY	ORDER OF ROTATIONAL SYMMETRY	SHAPE
Square	$4$ right angles, $4$ sides of equal length	$4$	$4$
Rectangle	$4$ right-angles	$2$	$2$
Rhombus	$4$ sides of equal length, $2$ pairs of equal angles	$2$	$2$
Parallelogram	$2$ pairs of parallel equal sides, $2$ pairs of equal angles	$0$	$2$
Trapezium	$1$ pair of parallel sides	$0$ , or $1$ in the case of an isoceles trapezium	$0$
Kite	$2$ pairs of equal sides, $1$ pair of equal angles	$1$	$0$

Interior and exterior angles

An interior angle of a polygon is the inner angle formed by two sides of the shape.

An exterior angle of a polygon is the angle formed by extending one side of the shape past the vertex.

Finding the interior angle of a regular polygon

The sum of the interior angles of a polygon is found by using the formula:

$\theta =(n-2)\times180$

Where $\theta$ is the sum of the interior angles, and $n$ is the number of sides.

Example 1

What is the size of an interior angle of a regular nonagon?

A nonagon has $9$ sides, so $n=9$ . The sum of the interior angles is therefore given to be:

$\theta=(9-2)\times180=7\times180$

$\theta=1260^\circ$

To find the size of one interior angle, divide this number by $9$ :

$1260\div9=140$

The size of one interior angle in a nonagon is $\underline{140^\circ}$ .

Finding the exterior angle of a regular polygon

Exterior angles of a regular polygon always add up to $360^\circ$ . Therefore, the formula for one exterior angle is given to be:

$\theta=\frac{360}{n}$

Where $\theta$ is the size of one exterior angle, and $n$ is the number of sides.

Example 2

What is the size of one exterior angle of a nonagon?

As in the example above, $n=9$ . Hence:

$\theta=\frac{360}{9}=40$ 

The size of one exterior angle of a nonagon is $\underline{140^\circ}$ .

Note: Notice how the exterior and interior angle of a regular polygon add up to $180^\circ$ . This is always true because both angles are on the same straight line, and the angles in a straight line add up to $180^\circ$ .

Learn with Basics

Length:

Unit 1

2D shapes: Types and properties

Jump Ahead

Unit 2