Angles around a point | Year 6

01 / LEARN

Understand the idea.

A full turn

One full turn around a point measures $360^{\circ}$ . The angles total $360^{\circ}$ when they fill the entire turn without gaps or overlaps. Trace around the vertex until you return to the starting arm to identify the whole.

Add the known parts of that turn and subtract their total from $360^{\circ}$ . Count every angle once. Before calculating, decide whether the diagram shows a full turn or only a straight half-turn; the totals are different.

Equal unknown parts

If the remaining space contains equal angles, subtraction finds their combined size, not the size of just one. Divide that remainder by the number of equal parts only when the question or markings tell you they are equal.

For a full turn containing $80^{\circ}$ and two equal angles, the remainder is $360^{\circ} - 80^{\circ} = 280^{\circ}$ . Each equal angle is $280^{\circ} \div 2 = 140^{\circ}$ . Checking all three parts confirms $80^{\circ} + 140^{\circ} + 140^{\circ} = 360^{\circ}$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

One missing sector

Three angles around a point are 70°, 120° and x. Find x.

Worked solution

1
Add the known angles.
$70^{\circ} + 120^{\circ} = 190^{\circ}$
These two parts already occupy 190 degrees of the full turn.
2
Subtract from a full turn.
$x = 360^{\circ} - 190^{\circ} = 170^{\circ}$
The three angles together fill the entire space around the point.
3
Check the total around the point.
$70^{\circ} + 120^{\circ} + 170^{\circ} = 360^{\circ}$
The result completes exactly one turn.

Answer

170°

EXAMPLE 02

Four parts

Four angles fill a full turn. Three are 90°, 85° and 110°. Find the fourth.

Worked solution

1
Combine all three given angles.
$90^{\circ} + 85^{\circ} + 110^{\circ} = 285^{\circ}$
All three known parts belong to the same full turn.
2
Find the fourth angle.
$x = 360^{\circ} - 285^{\circ} = 75^{\circ}$
The missing angle is the part of the full turn not yet counted.
3
Check the four parts.
$90^{\circ} + 85^{\circ} + 110^{\circ} + 75^{\circ} = 360^{\circ}$
The four angles fill the full turn with no remainder.

Answer

75°

03 / YOUR TURN

Try it yourself.

Check your understanding

Angles around a point are 100°, 150° and x. Find x.

Show answer & reasoning

110°

The known angles total $100^{\circ} + 150^{\circ} = 250^{\circ}$ . The remaining part of the full turn is $360^{\circ} - 250^{\circ} = 110^{\circ}$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Three angles around a point are $90^{\circ}$ , $130^{\circ}$ and $x$ . Find $x$ .
2 marks
Four angles around a point are $65^{\circ}$ , $95^{\circ}$ , $110^{\circ}$ and $x$ . Find $x$ .
2 marks
One angle is $120^{\circ}$ . The other two angles around the point are equal. Find each equal angle.
2 marks
Six equal angles fill a full turn. Find each angle.
1 mark
Mia lists angles of $90^{\circ}$ , $90^{\circ}$ , $80^{\circ}$ and $90^{\circ}$ for one complete turn. What is wrong?
2 marks
Three angles around a point are $75^{\circ}$ , $140^{\circ}$ and $x$ . Draw a rough sketch, find $x$ and check the total.
3 marks

Read the worked solutions

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

Three angles around a point are $90^{\circ}$ , $130^{\circ}$ and $x$ . Find $x$ .
2 marks
$360^{\circ} - (90^{\circ} + 130^{\circ}) = 140^{\circ}$ .
$140^{\circ}$
Four angles around a point are $65^{\circ}$ , $95^{\circ}$ , $110^{\circ}$ and $x$ . Find $x$ .
2 marks
$65 + 95 + 110 = 270$ .
$360 - 270 = 90$ .
$90^{\circ}$
One angle is $120^{\circ}$ . The other two angles around the point are equal. Find each equal angle.
2 marks
$360^{\circ} - 120^{\circ} = 240^{\circ}$ .
$240^{\circ} \div 2 = 120^{\circ}$ .
$120^{\circ}$
Six equal angles fill a full turn. Find each angle.
1 mark
$360^{\circ} \div 6 = 60^{\circ}$ .
$60^{\circ}$
Mia lists angles of $90^{\circ}$ , $90^{\circ}$ , $80^{\circ}$ and $90^{\circ}$ for one complete turn. What is wrong?
2 marks
$90 + 90 + 80 + 90 = 350$ .
A full turn is $360^{\circ}$ , leaving a $10^{\circ}$ gap.
They total $350^{\circ}$ , so $10^{\circ}$ is missing.
Three angles around a point are $75^{\circ}$ , $140^{\circ}$ and $x$ . Draw a rough sketch, find $x$ and check the total.
3 marks
$75^{\circ} + 140^{\circ} = 215^{\circ}$ .
$360^{\circ} - 215^{\circ} = 145^{\circ}$ .
$75^{\circ} + 140^{\circ} + 145^{\circ} = 360^{\circ}$ .
$x = 145^{\circ}$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 6 angle relationships: straight-line angles, angles around a point, vertically opposite angles and connected reasoning.

Official curriculum source ↗

In this lesson

Understand the idea.

A full turn

Equal unknown parts

See the method in action.

One missing sector

Add the known angles.

Subtract from a full turn.

Check the total around the point.

Four parts

Combine all three given angles.

Find the fourth angle.

Check the four parts.

Try it yourself.

Check your understanding

Keep practising