Direction Sense: Turns, Distance and the Compass

By the end of this lesson you will be able to follow a person through a chain of walks and turns, say which direction they end up facing, work out how far they are from where they began, and read a shadow to find a direction.

Already confident? Skip to practice: Direction Sense quiz

Before this: keep the right-angled triangle in mind. When someone walks east and then north, the two legs meet at a right angle, so the straight-line distance back to the start is the hypotenuse: distance = sqrt(east leg squared + north leg squared). A few clean triples make the arithmetic painless — 3, 4, 5 and its doubles (6, 8, 10 and 9, 12, 15), 5, 12, 13, and 8, 15, 17. Learn those four and most distance questions become one line of work.

What a direction-sense question looks like

You are given a short story: a person starts somewhere, walks a few distances, turns left or right between them, and you are asked one of three things.

• How far, and in which direction, from the start? You add up the legs and, if the path bends, use Pythagoras.
• Which direction is she facing now? You track the turns only; the distances do not matter.
• A shadow question. You are told where a shadow falls at sunrise or sunset and asked which way a person faces.

Every one of them rests on the same compass. Get the compass fixed in your head and the rest is bookkeeping.

The compass, and what a turn does

The four main directions, going clockwise, are North, East, South, West — remember the order by the phrase "Naughty Elephants Squirt Water". Between each pair sits a diagonal: North-East (between N and E), South-East, South-West and North-West. If you end up both north and east of the start, your direction from the start is North-East; both south and west, South-West; and so on.

A right turn is clockwise, a left turn is anticlockwise. The single most useful fact in the whole topic is what a right turn does to the direction you face:

• Facing North, a right turn makes you face East.
• Facing East, a right turn makes you face South.
• Facing South, a right turn makes you face West.
• Facing West, a right turn makes you face North.

A left turn runs the other way: North to West to South to East and back to North. A 180-degree turn (an about-turn) swaps a direction for its opposite: North for South, East for West.

The trap hidden here is that a turn is always relative to the direction you are facing right now, not to the compass. "Turn right" does not mean "go east". If you are already facing south, turning right makes you face west. Always ask "which way am I facing?" before you apply a turn.

Tracking the facing direction

For a "which direction is she facing now?" question, ignore the distances completely and walk the turns one at a time, writing the new facing after each:

1. Write the starting direction.
2. For each turn, slide one step around the compass — clockwise for right, anticlockwise for left, two steps for 180 degrees.
3. The last direction you write is the answer.

Example 1: counting the turns. A woman is facing South. She turns left, then left again, then right. Which way does she face?

1. Start: South.
2. Left (anticlockwise) from South goes to East.
3. Left again from East goes to North.
4. Right (clockwise) from North goes to East.

She is facing East. Notice that two different turn sequences can land on the same direction, so there is no shortcut — track them in order.

A turn need not be 90 degrees. A 45-degree turn lands you on a diagonal: a right turn of 45 degrees from North faces you North-East, and a further 45 degrees faces you East. It helps to measure angles clockwise from North — North is 0, North-East is 45, East is 90, South-East is 135, South is 180, and so on round to North-West at 315. Then any clockwise turn is just addition (subtract for an anticlockwise turn), and you read the total back off the circle.

Net displacement: adding up the legs

For a "how far from the start?" question, keep two running totals: one for the east-west line and one for the north-south line. Treat east and north as adding, west and south as subtracting (or just keep two separate piles and subtract at the end).

1. Set east-west = 0 and north-south = 0.
2. For each leg, find the direction the person is actually moving, and add that distance to the matching total.
3. At the end, the two totals give the net east-west and net north-south distance. Opposite legs cancel; this is why a path with four turns can leave someone only a short way from home.

Example 2: legs that partly cancel. A man walks 15 m north, turns right and walks 9 m, turns right and walks 7 m, turns right and walks 9 m. How far from the start, and in which direction?

1. North 15 — north-south = 15 (north), facing north.
2. Right turn, now facing east, walk 9 — east-west = 9 (east).
3. Right turn, now facing south, walk 7 — north-south = 15 - 7 = 8 (north).
4. Right turn, now facing west, walk 9 — east-west = 9 - 9 = 0.

The east and west legs cancel exactly. The net movement is 8 m north and nothing east or west, so he is 8 m from the start, due North. The distances 9 and 9 were there only to be cancelled — a common way to dress up an easy answer.

The working is easiest to keep straight if you lay it out in two columns as you read, one headed east-west and one headed north-south, and jot each leg under the right column with a plus or minus sign. When the story ends, each column has a single signed total, and you are ready for step 5 or 6 of the method below.

Distance when the path bends: Pythagoras

When the net east-west total and the net north-south total are both non-zero, the person is neither straight along a line from you nor on a diagonal you can read off — they are at the corner of a right-angled triangle. The straight-line ("as the crow flies") distance is the hypotenuse:

distance = sqrt( (net east-west)^2 + (net north-south)^2 )

This is where the clean triples earn their keep.

Example 3: a single bend. A boy walks 9 km east, then turns left and walks 12 km. How far is he from the start, and in which direction?

1. East 9 — east-west = 9 (east), facing east.
2. Left turn, now facing north, walk 12 — north-south = 12 (north).
3. Both totals are non-zero: distance = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 km.
4. He is east and north of the start, so the direction is North-East.

That is the 9, 12, 15 triple (which is just 3, 4, 5 times three). The answer is 15 km, North-East.

Example 4: several legs, then Pythagoras. A girl walks 8 km north, then 4 km east, then 8 km north, then 4 km east. How far from the start?

1. Add by direction. North: 8 + 8 = 16 km. East: 4 + 4 = 8 km.
2. Distance = sqrt(16^2 + 8^2) = sqrt(256 + 64) = sqrt(320), which is not a whole number.

So pause — this shows the method, not a tidy triple. When the totals are 16 and 8 the honest answer is sqrt(320) which is about 17.9 km; in a real exam the numbers are chosen so the triple is clean. Keep the method, and expect the setter to hand you a 5, 12, 13 or an 8, 15, 17 rather than an ugly root. Your turn: redo it with the girl walking 15 km north in total and 8 km east in total. What is the distance?

Check: sqrt(15^2 + 8^2) = sqrt(225 + 64) = sqrt(289) = 17 km (the 8, 15, 17 triple), and she is North-East of the start.

Example 5: a longer chain. A man walks 20 m west, turns left and walks 15 m, turns left and walks 15 m, turns right and walks 9 m. Where is he relative to the start?

1. West 20 — east-west = 20 (west), facing west.
2. Left turn, now facing south, walk 15 — north-south = 15 (south).
3. Left turn, now facing east, walk 15 — east-west = 20 - 15 = 5 (west).
4. Right turn, now facing south, walk 9 — north-south = 15 + 9 = 24 (south).
5. Net: 5 west and 24 south. Hmm — 5 and 24 is not one of our four triples, but 5, 24 belongs to the wider family: sqrt(5^2 + 24^2) = sqrt(25 + 576) = sqrt(601), not clean.

Your turn: the setter would trim leg 4 so the south total is 12, giving net 5 west and 12 south. What is the straight-line distance and the direction?

Check: sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 m, and the man is west and south of the start, so South-West.

Shadows: the sunrise-west, sunset-east rule

The sun rises in the east and sets in the west. A shadow always points away from the sun, so:

• At sunrise (early morning) the sun is in the east, and every shadow falls towards the west.
• At sunset (evening) the sun is in the west, and every shadow falls towards the east.
• At noon the sun is nearly overhead, so a shadow is very short and gives no reliable direction. Treat any "midday shadow points somewhere" claim with caution — the standard exam answer is that there is effectively no usable shadow.

To turn that into a facing direction, pin down where the shadow sits relative to the person and reason from the body.

Example 6: a morning shadow. At sunrise, a pole's shadow falls straight ahead of a boy standing beside it, pointing in the direction he faces. Which way is he facing?

1. Sunrise, so the shadow points west.
2. The shadow is straight ahead of him, so west is straight ahead.
3. He is facing West.

Example 7: shadow to one side. One evening at sunset, a man finds his shadow falling to his left. Which direction is he facing?

1. Sunset, so the shadow points east.
2. The shadow is on his left, so east is on his left.
3. For a person whose left hand points east, the front points south (stand facing south: east is on your left, west on your right).
4. He is facing South.

The reliable move is to fix the shadow's compass direction first (west at sunrise, east at sunset), then rotate your body in your head until that direction sits where the question puts the shadow.

How to attack any direction-sense question

1. Decide which kind it is: facing-direction only, distance-and-direction, or shadow.
2. For a facing question, write the start direction and step around the compass once per turn (clockwise for right, anticlockwise for left, two steps for 180). The last direction is the answer.
3. For a distance question, keep two totals — east-west and north-south. For each leg, work out the direction actually moved (apply the turn to the current facing first), then add to the right total.
4. Let opposite legs cancel. East cancels west, north cancels south.
5. If only one total is left, the answer is that distance in that single direction (due North, due East, and so on).
6. If both totals remain, use distance = sqrt(east-west^2 + north-south^2), and name the diagonal (North-East, South-West, etc.) from the two surviving directions.
7. For a shadow, fix the shadow's direction first — west at sunrise, east at sunset — then rotate your body to match where the shadow is placed.
8. Always re-check each turn against the current facing, never against the compass.

At a glance: the rules in one place

Learn these by heart: the clockwise order North, East, South, West; that a right turn is clockwise and a left turn anticlockwise; the four clean triples 3-4-5 (with 6-8-10 and 9-12-15), 5-12-13 and 8-15-17; and the shadow rule — sunrise shadow west, sunset shadow east.

• Right turn (clockwise): N to E to S to W to N. Left turn (anticlockwise): N to W to S to E to N. A turn is relative to your current facing, not the compass.
• 180-degree turn: swap for the opposite direction (N and S, E and W).
• Net displacement: add east and north, subtract west and south; opposite legs cancel.
• One total left over: the answer is that distance, due that single direction.
• Both totals left over: distance = sqrt(east-west^2 + north-south^2); the direction is the diagonal named by the two surviving directions.
• Shadow: sun rises east so morning shadows fall west; sun sets west so evening shadows fall east; at noon there is no usable shadow. Fix the shadow's direction, then rotate the body to place it.
• Direction FACED is not direction MOVED: after a turn you may face one way but, on the next instruction, that is the way you then move — keep the two ideas separate.

Common mistakes

• Confusing the direction FACED with the direction MOVED. You track facing to decide which way the next walk goes, but the distance totals use the direction moved. Keep a note of the current facing beside the running totals.
• Turning relative to the compass instead of the current facing. "Turn right" while facing south is west, not east. Re-read your current facing before every turn.
• Swapping left and right. Right is clockwise (N to E); left is anticlockwise (N to W). Fix one of them firmly and the other is its reverse.
• Adding the legs instead of using Pythagoras. When the path bends, 9 km then 12 km is not 21 km from the start — it is sqrt(9^2 + 12^2) = 15 km. The sum of the legs is always too big.
• Forgetting that opposite legs cancel. A 9 m east followed later by a 9 m west leaves you 0 east-west, not 18.
• Mixing up the shadow times. Morning shadows fall west (sun in the east); evening shadows fall east (sun in the west). Reversing them flips every answer.

Next

The remaining Reasoning topics include Puzzles and Seating Arrangement, Blood Relations and Non-Verbal Reasoning. The right-triangle arithmetic you used here is the same you will meet in Mensuration on the Quantitative Aptitude side, so the triples are worth memorising once for both.

Ready to test this lesson? Take the Direction Sense quiz

How this was made: Drafted with AI assistance; every formula and worked figure recomputed independently; spot-checked by the editor on 30 October 2026.