Polygon angle sums | Year 7

01 / LEARN

The idea, step by step

Count the triangles, not the corners

A polygon is a closed shape made from straight sides. A convex polygon has no inward-pointing corner. Draw diagonals from one vertex of a convex n-sided polygon: it divides into $n - 2$ triangles without gaps or overlap. Each triangle contributes $180^{\circ}$ .

Therefore a simple n-sided polygon has interior-angle total $(n - 2) \times 180^{\circ}$ . For example, a pentagon divides into three triangles, so its total is $540^{\circ}$ . This is the total for all corners, not one angle.

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

Regular means equal sides and equal angles

A regular polygon has all sides equal and all interior angles equal. Divide the interior-angle total by n only when the angles are known to be equal. In an irregular polygon the total still works, but individual angles may differ.

For a convex polygon, turning through one exterior angle at each vertex completes a full turn of $360^{\circ}$ . A regular polygon therefore has exterior angle $360^{\circ} \div n$ . Its interior and adjacent exterior angle add to $180^{\circ}$ .

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

A common trap

Divide the polygon angle total by the number of sides for every polygon.

Instead: Division gives the size of each angle only when they are all equal, as in a regular polygon. It gives only their average otherwise.

02 / WORKED EXAMPLES

See the thinking

Example 1

A regular pentagon

Find each interior angle of a regular pentagon.

There are five sides, so the total is $(5 - 2) \times 180^{\circ} = 540^{\circ}$ .
All five interior angles are equal.
$540 \div 5 = 108$ .

Each interior angle is

108^{\circ}

Example 2

An irregular hexagon

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

A hexagon has total $(6 - 2) \times 180^{\circ} = 720^{\circ}$ .
The known angles total $110 + 120 + 130 + 105 + 140 = 605$ degrees.
$720 - 605 = 115$ .

The missing angle is

115^{\circ}

Check your understanding

Find each interior angle of a regular hexagon.

Show answer

$120^{\circ}$ .

The total is $(6 - 2) \times 180 = 720$ degrees. Divide by six: $720 \div 6 = 120$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF 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 ↗

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