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.

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 $$180^\circ$$.

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