Polygon angle sums | Year 7

01 / LEARN

Understand the idea.

Count the triangles, not the corners

A polygon is a closed shape formed from straight sides. For a convex polygon, diagonals from one vertex divide it into $n - 2$ triangles without overlap. Their angle sums combine to give the polygon total $(n - 2) \times 180^{\circ}$ .

This total also holds for a simple concave polygon, though a different triangulation may be needed. The formula assumes a flat polygon whose boundary does not cross itself. It gives the total of all interior angles, not the size of each corner.

S = (n - 2) \times 180^{\circ}

Regular means equal sides and equal angles

A regular polygon has equal sides and equal interior angles. Only when the angles are equal can you divide their total by the number of sides to find each one. For an irregular polygon, use any given angle information instead.

For a convex polygon, one exterior turn at each vertex totals $360^{\circ}$ . In a regular polygon these turns are equal, so each is $\frac{360^{\circ}}{n}$ . An interior angle and its adjacent exterior angle form a straight angle.

I = \frac{(n - 2) 180^{\circ}}{n}, E = \frac{360^{\circ}}{n}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A regular pentagon

Find each interior angle of a regular pentagon.

Worked solution

1
Find the pentagon interior-angle total.
$S = (5 - 2) \times 180^{\circ} = 540^{\circ}$
A pentagon has 5 sides and can be divided into 3 triangles.
2
Divide the total among equal angles.
$I = \frac{540^{\circ}}{5} = 108^{\circ}$
Regularity means all 5 interior angles have the same size.
3
Check the total.
$5 \times 108^{\circ} = 540^{\circ}$
Five equal angles of 108 degrees reproduce the pentagon angle sum.

Answer

Each interior angle is $108^{\circ}$ .

EXAMPLE 02

An irregular hexagon

Five interior angles of a hexagon are 110, 120, 130, 105 and 140 degrees. Find the sixth.

Worked solution

1
Find the hexagon interior-angle total.
$S = (6 - 2) \times 180^{\circ} = 720^{\circ}$
A simple 6-sided polygon has the angle sum of 4 triangles.
2
Add the five supplied angles.
$\begin{aligned} 110^{\circ} + 120^{\circ} + 130^{\circ} & = 360^{\circ} \\ 360^{\circ} + 105^{\circ} + 140^{\circ} & = 605^{\circ} \end{aligned}$
These are the known contributions to the total.
3
Subtract to find the missing angle.
$x = 720^{\circ} - 605^{\circ} = 115^{\circ}$
The sixth angle must supply the remaining part of the hexagon sum.
4
Check the completed sum.
$605^{\circ} + 115^{\circ} = 720^{\circ}$
The six angles now have the correct combined total.

Answer

The missing angle is $115^{\circ}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Find each interior angle of a regular hexagon.

Show answer & reasoning

$120^{\circ}$ .

A hexagon has interior-angle total $(6 - 2) \times 180^{\circ} = 720^{\circ}$ . Regularity makes the six angles equal, so each is $\frac{720^{\circ}}{6} = 120^{\circ}$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Find the interior-angle sum of a quadrilateral.
2 marks
Find each interior angle of a regular octagon.
2 marks
Find the interior-angle sum of a heptagon, which has seven sides.
2 marks
A simple polygon has interior-angle sum $1440^{\circ}$ . How many sides does it have?
3 marks
A regular polygon has exterior angle $24^{\circ}$ . Find its number of sides and each interior angle.
3 marks
Four interior angles of a pentagon are 100, 115, 90 and 120 degrees. Find the fifth.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the interior-angle sum of a quadrilateral.
2 marks
$(4 - 2) \times 180^{\circ} = 360^{\circ}$ .
$360^{\circ}$ .
Find each interior angle of a regular octagon.
2 marks
The total is $(8 - 2) \times 180 = 1080$ degrees.
$1080 \div 8 = 135$ .
$135^{\circ}$ .
Find the interior-angle sum of a heptagon, which has seven sides.
2 marks
$(7 - 2) \times 180^{\circ} = 900^{\circ}$ .
$900^{\circ}$ .
A simple polygon has interior-angle sum $1440^{\circ}$ . How many sides does it have?
3 marks
$(n - 2) 180 = 1440$ .
$n - 2 = 8$ , so $n = 10$ .
10 sides.
A regular polygon has exterior angle $24^{\circ}$ . Find its number of sides and each interior angle.
3 marks
$360 \div 24 = 15$ .
The adjacent interior angle is $180 - 24 = 156$ degrees.
15 sides; interior angle $156^{\circ}$ .
Four interior angles of a pentagon are 100, 115, 90 and 120 degrees. Find the fifth.
3 marks
The total is $540^{\circ}$ .
The four given angles total $425^{\circ}$ .
$540 - 425 = 115$ degrees.
$115^{\circ}$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Mathematics

Year 7 angle relationships, polygon properties and classification algorithms. Chapter grouping follows the approved Ascent outline.

Official curriculum source ↗

In this lesson

Understand the idea.

Count the triangles, not the corners

Regular means equal sides and equal angles

See the method in action.

A regular pentagon

Find the pentagon interior-angle total.

Divide the total among equal angles.

Check the total.

An irregular hexagon

Find the hexagon interior-angle total.

Add the five supplied angles.

Subtract to find the missing angle.

Check the completed sum.

Try it yourself.

Check your understanding

Keep practising