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Year 6 Maths Angles

Angles at a point

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Theory

Angles that meet at a point and fill the space all the way round make a full turn, so they add to \(360^\circ\). This Year 6 maths page shows how to find unknown angles at a point, including when there are right angles, four or five angles, or equal angles, and how to check whether a set of angles can really meet at a point.

A full turn (a revolution) is \(360^\circ\). It is \(4\) right angles, or \(2\) straight angles.

Angles at a point share one vertex and together go all the way round it with no gaps or overlaps. They add to \(360^\circ\).

To find an unknown angle at a point, subtract the known angles from \(360^\circ\).

Equal angles at a point share the \(360^\circ\) equally: \(6\) equal angles are \(360 \div 6 = 60^\circ\) each.

Three angles at a point Three rays from one point make three angles: 135 degrees, 105 degrees and an unknown angle, filling the space all the way round. 135° 105° ?
\(360 - 135 - 105 = 120^\circ\).
A right angle at a point Three rays from one point: a right angle marked with a square, an angle of 145 degrees and an unknown angle. 145° ?
With a right angle: \(360 - 90 - 145 = 125^\circ\).
Four angles at a point Four rays from one point make four angles: 80, 115 and 95 degrees and an unknown angle. 80° 115° 95° ?
Four angles: \(360 - (80 + 115 + 95) = 70^\circ\).

The full-turn fact:

\[ a + b + c + \ldots = 360^\circ \]
a + b + c +=360
\[ \text{unknown} = 360^\circ - \text{known angles} \]
unknown =360− known angles
\[ 360^\circ = 4 \times 90^\circ = 2 \times 180^\circ \]
360=4×90=2×180
360, not 180. Angles all the way round a point make a full turn: \(360^\circ\).
Add first, then subtract once. Add the known angles, then take the total from \(360^\circ\).

Finding an unknown angle at a point

  1. Check that the angles go all the way round one point.
  2. Add the known angles (right-angle marks are \(90^\circ\)).
  3. Subtract from \(360^\circ\).
  4. Check that all the angles add to \(360^\circ\).
Example 1 — Three angles
Use the first diagram. Find the missing angle.
Solution
\(135 + 105\)\(=\)\(240\)
\(360 - 240\)\(=\)\(120\)
360−240=120

\(120^\circ\).

Example 2 — With a right angle
Use the second diagram. Find the missing angle.
Solution
\(90 + 145\)\(=\)\(235\)
\(360 - 235\)\(=\)\(125\)
360−235=125

\(125^\circ\).

Example 3 — Four angles
Use the third diagram. Find the missing angle.
Solution
\(80 + 115 + 95\)\(=\)\(290\)
\(360 - 290\)\(=\)\(70\)
360−290=70

\(70^\circ\).

Example 4 — Three sets
Each set meets at a point with one more angle. Find the missing angle.
(i) \(235^\circ\)
(ii) \(170^\circ\), \(90^\circ\)
(iii) \(100^\circ\), \(85^\circ\), \(65^\circ\)
Solution
\(360 - 235\)\(=\)\(125\)
\(360 - 260\)\(=\)\(100\)
\(360 - 250\)\(=\)\(110\)
125,100,110

(i) \(125^\circ\)
(ii) \(100^\circ\)
(iii) \(110^\circ\)

Example 5 — Five angles
Five angles meet at a point. Four are \(55^\circ\), \(80^\circ\), \(65^\circ\) and \(95^\circ\). Find the fifth.
Solution
\(55 + 80 + 65 + 95\)\(=\)\(295\)
\(360 - 295\)\(=\)\(65\)
360−295=65

\(65^\circ\).

Example 6 — Equal angles
(i) Three equal angles meet at a point. How big is each?
(ii) One angle at a point is \(150^\circ\) and the other two are equal. How big is each?
Solution
\(360 \div 3\)\(=\)\(120\)
\((360 - 150) \div 2\)\(=\)\(105\)
120,105

(i) \(120^\circ\)
(ii) \(105^\circ\)

Example 7 — Pie slices
A round pie is cut into three slices from the centre. Two slices have angles of \(125^\circ\) and \(140^\circ\). What is the angle of the third slice?
Solution
\(360 - 125 - 140\)\(=\)\(95\)
360−125−140=95

\(95^\circ\).

Example 8 — Possible or not?
Can angles of \(110^\circ\), \(95^\circ\), \(80^\circ\) and \(70^\circ\) meet at a point?
Solution
\(110 + 95 + 80 + 70\)\(=\)\(355\)
110+95+80+70=355

No. They add to \(355^\circ\), \(5^\circ\) short of a full turn.

Example 9 — Find the mistake
Four angles meet at a point: \(130^\circ\), \(75^\circ\), \(90^\circ\) and one unknown. Omar used \(180^\circ\) and got a negative answer. What is right?
Solution
\(130 + 75 + 90\)\(=\)\(295\)
\(360 - 295\)\(=\)\(65\)
360−295=65

Angles at a point add to \(360^\circ\), not \(180^\circ\). The unknown angle is \(65^\circ\).

Common pitfalls

Using 180°. That is only a straight line. A full turn is \(360^\circ\).
Missing a right angle. A square mark is \(90^\circ\) even with no number.
Adding wrongly. Add the known angles carefully, then subtract once.
Not checking. All the angles round the point must make exactly 360°.

Frequently asked questions

What do angles at a point add to?

\(360^\circ\), a full turn.

How do I find a missing angle at a point?

Add the known angles and subtract the total from 360°.

How big is each of 5 equal angles at a point?

\(360 \div 5 = 72^\circ\).

What is the difference between angles on a line and angles at a point?

On a line they add to \(180^\circ\); all the way round a point they add to \(360^\circ\).

Can 110°, 95°, 80° and 70° meet at a point?

No: they add to \(355^\circ\).

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