A full turn is 360°
An angle is made by two rays from one point. Hold one ray still and turn the other around the point, and the angle grows as it turns. A quarter turn is a right angle, 90°. A half turn is a straight line, 180°. Keep turning until the ray is back where it started, and it has made one full turn. A full turn is 360°.
So a ray swung round by 150° has made 150° of the full turn, and the rest of the turn back to its starting line is 360 − 150 = 210°.
Angles at a point add to 360°
Now draw three rays from the same point. They cut the full turn into three angles that sit side by side, with no gaps between them and no overlaps. Say the first angle is 150°. A second angle beside it takes another 120° of the turn, and a third angle of 90° closes it.
Together the three angles make the whole turn: 150 + 120 + 90 = 360. The same is true for any number of angles around a point, however many rays there are. Angles at a point add up to 360°.
Three rays from the center cut the full turn into angles of 150°, 120° and 90°. They fill it exactly: 150 + 120 + 90 = 360.
Finding a missing angle
When one angle at a point is unknown, subtract the others from 360. Two of the three angles at a point are 150° and 120°, so the third is 360 − 150 − 120 = 90°. Check by adding all three: 150 + 120 + 90 = 360.
It works the same with more angles. Four angles meet at a point, and three of them are 100°, 85° and 110°. The fourth is 360 − 100 − 85 − 110 = 65°.
When the missing angles are equal, subtract first and then share. Three equal angles meet at a point beside an angle of 60°. Together they make 360 − 60 = 300°, so each one is 300 ÷ 3 = 100°. Check: 100 + 100 + 100 + 60 = 360.
More than a half turn: reflex
An angle can be bigger than a straight line. A ray that turns 250° from its starting line has gone past the half turn, the bigger way round. An angle between 180° and 360° is called a reflex angle.
Two rays from one point always make two angles, one each way round, and together those two angles make the full turn. If the smaller one is 110°, the reflex one is 360 − 110 = 250°.
So every angle has a name by its size: acute is less than 90°, a right angle is exactly 90°, obtuse is between 90° and 180°, a straight angle is exactly 180°, and reflex is between 180° and 360°.
an angle is the turn between two arms; scaling both arms by any factor leaves θ unchanged
Turn the arm to 150°
Turn the arm to 150°, the first angle above. Keep turning past 180° and the name changes to reflex. At 360° the arm is back on its starting line: one full turn.
The usual mistakes
Using 180 in place of 360. A straight line is only a half turn, and angles all the way around a point make the full turn. With angles of 120° and 110° at a point, the third is 360 − 120 − 110 = 130°. Working with 180 gives 120 + 110 − 180 = 50°, which is only how far the two angles together go past a straight line.
Using 270. Three quarter turns make 270°, but a full turn is four quarter turns, 360°.
Calling a reflex angle obtuse. An obtuse angle is less than 180°. An angle of 250° has gone past the straight line, so it is reflex.
Naming an angle with three letters
When several rays meet at a point O, an angle is named by three letters. ∠AOB is the angle at O between the rays OA and OB. The middle letter is always the point where the arms meet.
Worked example: Angles at a Point
Question Four rays OA, OB, OC and OD meet at O. ∠ AOB = 90° and ∠ BOC = 115°. ∠ DOA is 15° more than ∠ COD. Find ∠ COD and ∠ DOA.
1.The four angles around O make a full turn: they add to 360°.
Four rays from O. The right angle and the 115° are given; the other two differ by 15°. 2.The two known angles: ∠ AOB + ∠ BOC = 90° + 115° = 205°.
The known pair: 90° + 115° = 205°. 3.What is left for ∠ COD and ∠ DOA together: 360° − 205° = 155°.
The rest of the turn: 360° − 205° = 155° for ∠ COD and ∠ DOA together. 4.∠ DOA is 15° more than ∠ COD. Take that 15° off first: 155° − 15° = 140° is two equal shares.
Take off the 15° difference: 155° − 15° = 140°, two equal shares. 5.∠ COD = 140° ÷ 2 = 70°.
∠ COD = 140° ÷ 2 = 70°. 6.∠ DOA = 70° + 15° = 85°. Check: 90 + 115 + 70 + 85 = 360.
∠ DOA = 70° + 15° = 85°. The four angles add to 360°.
Answer: ∠ COD = 70°; ∠ DOA = 85°
Common mistakes
- Adding the angles to 180° as if O sat on a straight line; four rays from one point make a full turn.
- Halving 155° straight away and giving 77.5°: the two angles are not equal, one is 15° more.
Turning to face a compass direction
A turn is often measured from a compass direction. North, East, South and West are a quarter turn apart. Turning clockwise, the way the hands of a clock go round, from North: 90° faces East, 180° faces South, 270° faces West, and 360° faces North again. Turning the other way is counter-clockwise.
Halfway between each pair of neighbors are North-East, South-East, South-West and North-West, each 45° from the directions beside it. So a turn of 225° clockwise from North passes East at 90° and South at 180°, then goes 45° further, to South-West. A complete turn of 360° leaves a pointer facing the way it started.
A turn of 225° clockwise from North, past East and South, ends facing South-West.
Worked example: A Pointer Turned Both Ways, Several Times Round
Question A surveying pointer starts facing North. It makes 3 complete turns clockwise, then turns 225° clockwise, and finally turns counter-clockwise through 112 right angles. (a) In which compass direction does the pointer face at the end? (b) The net clockwise rotation is the angle turned clockwise minus the angle turned counter-clockwise. Find the pointer's net clockwise rotation in degrees. (c) The pointer is then fixed to an arm that turns clockwise at 0.5° per second from 09:15 to 10:55. Find the total angle the arm turns through, in right angles.
1.Three complete turns are 3 × 360° = 1080° clockwise, and after each whole turn the pointer faces North again.
Three complete turns: 3 × 360° = 1080°, and the pointer faces North again. 2.Turning 225° clockwise from North passes East at 90° and South at 180°, then goes 45° further: the pointer faces South-West.
225° clockwise passes East and South and goes 45° further: South-West. 3.112 right angles are 112 × 90° = 135°. Turning 135° counter-clockwise from South-West takes 45° back to South and 90° more to East. (a) The pointer faces East.
(a) 112 right angles = 135° back: 45° to South, 90° more to East. 4.(b) Clockwise the pointer turned 1080° + 225° = 1305°, so the net clockwise rotation is 1305° − 135° = 1170°. Check: 1170° = 3 × 360° + 90°, a quarter turn clockwise past North, which is East.
(b) Net clockwise rotation: 1080° + 225° − 135° = 1170°. 5.From 09:15 to 10:55 is 1 h 40 min, which is 100 min, or 100 × 60 = 6000 s.
From 09:15 to 10:55 is 100 minutes, 6000 seconds. 6.(c) The arm turns 6000 × 0.5° = 3000°, and 3000 ÷ 90 = 3313, so it turns through 3313 right angles.
(c) 6000 × 0.5° = 3000°, and 3000 ÷ 90 = 3313 right angles.
Answer: (a) East (b) 1170° (c) 3313 right angles
Common mistakes
- Adding the counter-clockwise turn to the clockwise ones and answering 1080° + 225° + 135° = 1440°: a turn the other way undoes part of the clockwise turning, so it is taken away.
- Leaving out the three complete turns in (b) and answering 90°: they do not change the direction, but the pointer still turned through them, and the net rotation counts every degree.