Angles and Symmetry
Angle Symmetry in Regular Polygons

Regular polygons will have both line and rotational symmetry. This symmetry can be used to find the interior angles of a regular polygon.

Worked Examples

1

Find the interior angle of a regular dodecagon.

Show me

The diagram shows how a regular dodecagon can be split into 12 isosceles triangles.

As there are 360° around the centre of the dodecagon, the centre angle in each triangle is

= 30°

So the other angles of each triangle will together be

180° − 30° = 150°

Therefore each of the other angles will be

= 75°

As two adjacent angles are required to form each interior angle of the dodecagon, each interior angle will be

75° × 2 = 150°

As there are 12 interior angles, the sum of these angles will be 12 × 150° = 1800°.
2

Find the sum of the interior angles of a regular heptagon.

Show me

Split the heptagon into 7 isosceles triangles.

Each triangle contains three angles which add up to 180° , so the total of all the marked angles will be

7 × 180° = 1260°.

However the angles at the point where all the triangles meet should not be included, so the sum of the interior angles is given by

1260° − 360° = 900°
3
(a)

Copy the octagon shown in the diagram and draw in any lines of symmetry.

Show me

There is only one line of symmetry as shown in the diagram.
(b)

Copy the octagon and shade in extra triangles so that it now has rotational symmetry.

Show me

The original octagon has no rotational symmetry.

By shading the extra triangle shown, it has rotational symmetry of order 4.

By shading all the triangles, it has rotational symmetry of order 8.

Exercises

Find the interior angle for a regular:

(a)
pentagon °
(b)
hexagon °
(c)
octagon °
(d)
decagon (10 sides). °

Find the sum of the interior angles in each polygon shown below.

(a)
°
(b)
°

Which regular polygons have interior angles of:

(a)
90°
(b)
120°
(c)
108°
(d)
140°
(e)
60°
(f)
144°?
(a)

A polygon has 9 sides. What is the sum of the interior angles?

°
(b)

Complete the table below.

ShapeSum of interior angles
Triangle180°
Square°
Pentagon°
Hexagon720°
Heptagon°
Octagon°
(c)

Describe a rule that could be used to calculate the sum of the interior angles for a polygon with n sides.

(d)

Find the sum of the interior angles for a 14-sided polygon.

°
(e)

The sum of the interior angles of a polygon is 1260°. How many sides does the polygon have?

A regular octagon, drawn opposite, has eight sides. One side of the octagon has been extended to form angle p.

(a)

Work out the size of angle p.

p = °

(b)

Work out the size of angle q.

q = °

The diagram shows three identical rhombuses, P, Q and T.

(a)

What is the size of angle x?

°
(b)

Rhombus Q can be rotated onto rhombus T. State the angle of rotation.

°
(c)

Write down the order of rotational symmetry of

(i)
a rhombus
(ii)
a regular hexagon.
(d)

The given shape could also represent a three dimensional shape. What is this shape?

Investigation

How many squares are there in the given figure?