Construction of geometric figures (Grade 9) — Mathematics Grades 8–11

Using a compass and ruler to construct and bisect angles, and finding polygon angle sums.

Gr 9 Term 2 — Construction of geometric figuresTerm 2

Practise Construction of geometric figures in the app →

What gets asked

Bisect an angle of a triangle using a compass and ruler.
Construct an angle of 30, 45 or 60 degrees without a protractor.
Calculate the sum of the interior angles of a polygon.
Calculate the size of one interior angle of a regular polygon.

You must be able to

Use compass arcs from the vertex to bisect an angle accurately.
Build 60 degrees, 90 degrees and their combinations with a compass and ruler.
Use (n - 2) times 180 degrees for a polygon's interior angle sum.
Divide the interior angle sum by the number of sides for a regular polygon.

Traps that cost marks

Measuring an angle with a protractor and halving it, instead of constructing the bisector. A bisector must be constructed with compass arcs, not measured with a protractor.
Using a protractor to draw 30, 45 or 60 degrees instead of constructing it. These angles come from bisecting 60 degrees or 90 degrees, built with compass arcs only.
Multiplying the number of sides by 180 degrees without subtracting 2. The interior angle sum is (n - 2) times 180 degrees. Do not skip the minus 2.

Worked example

Calculate the sum of the interior angles of a polygon with 7 sides.

Use $(n - 2) \times 180°$ with $n = 7$ .
$(7 - 2) \times 180° = 5 \times 180°$ .
The interior angle sum is $900°$ .

Angles of a polygon

How many degrees all the corners of a straight-sided shape add up to, and where the minus two comes from.

polygon: A closed shape made of straight sides only. A triangle and a square are polygons.
regular polygon: A polygon with all its sides equal and all its angles equal.
interior angle: An angle inside the shape, at one of its corners.
diagonal: A straight line joining two corners that are not next to each other.

A polygon is closed and made only of straight sides. Count the sides and call that number $n$ . A shape with $n$ sides has $n$ corners, so it has $n$ interior angles.

Pick one corner and draw every diagonal you can from it. The polygon breaks into triangles. A shape with $n$ sides always breaks into $n - 2$ of them.

Every triangle holds $18 0^{\circ}$ , so the interior angles add up to $(n - 2) \times 18 0^{\circ}$ . The minus $2$ is the whole point of the formula. Leaving it out is the commonest mistake here.

Test the formula on a shape you already know. A quadrilateral has $4$ sides, and $(4 - 2) \times 180 = 360$ . That is the answer you expect, so the formula is behaving.

In a regular polygon every interior angle is the same size. To find ONE of them, work out the total first, then share it out. Divide the total by the number of angles.

Rules to remember

interior angles of a polygon: $(n - 2) \times 18 0^{\circ}$
one angle of a regular polygon: $\frac{( n - 2 ) \times 18 0 ^{\circ}}{n}$

Examples

Find the sum of the interior angles of a hexagon.

Worked answer

A hexagon has $6$ sides, so $n = 6$ .
$6 - 2 = 4$ , so it breaks into $4$ triangles.
$4 \times 180 = 720$ .
The interior angles add up to $72 0^{\circ}$ .

Answer: $72 0^{\circ}$

Forgetting the minus $2$ would give $108 0^{\circ}$ , which counts two triangles too many.

Find the size of ONE interior angle of a regular octagon.

Worked answer

An octagon has $8$ sides, so $n = 8$ .
Total: $(8 - 2) \times 180 = 1080$ .
A regular octagon has $8$ equal angles.
$1080 \div 8 = 135$ , so each angle is $13 5^{\circ}$ .

Answer: $13 5^{\circ}$

The question asked for one angle, so the total still has to be shared out.

The interior angles of a polygon add up to

144 0^{\circ}

. How many sides has it?

Worked answer

Start from $(n - 2) \times 180 = 1440$ .
Divide both sides by $180$ : $n - 2 = 8$ .
Add $2$ to both sides: $n = 10$ .
The polygon has $10$ sides.

Answer: $10$ sides

Running the formula backwards means undoing it one step at a time.

Traps

Multiplying the number of sides by 180 degrees without subtracting 2. The interior angle sum is (n - 2) times 180 degrees. Do not skip the minus 2.
Giving the total for all the corners when one angle of a regular polygon was asked for. Divide that total by the number of sides. A regular octagon gives $13 5^{\circ}$ , not $108 0^{\circ}$ .

Learn and practise “Angles of a polygon” in the app →

Grade 9 Construction of geometric figures

What gets asked

You must be able to

Traps that cost marks

Worked example

Angles of a polygon

Rules to remember

Examples

Traps

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