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∘ 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∘
3
3
Right-angled triangle
1 angle of 90∘
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:
θ=(n−2)×180
Where θ 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:
θ=(9−2)×180=7×180
θ=1260∘
To find the size of one interior angle, divide this number by 9:
1260÷9=140
The size of one interior angle in a nonagon is 140∘.
Finding the exterior angle of a regular polygon
Exterior angles of a regular polygon always add up to 360∘. Therefore, the formula for one exterior angle is given to be:
θ=n360
Where θ 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:
θ=9360=40
The size of one exterior angle of a nonagon is 140∘.
Note: Notice how the exterior and interior angle of a regular polygon add up to 180∘. This is always true because both angles are on the same straight line, and the angles in a straight line add up to 180∘.