Quantitative Aptitude · Direction Tests

Distance and Displacement in Direction Problems

Updated 1 October 2026 · Fact-checked

Distance is the total length of the path a person walks. Displacement is the straight-line gap between the start and the end point. To solve, draw the path, add all legs for distance, net out opposite legs along North-South and East-West, then apply Pythagoras for displacement.

Understand Distance and Displacement in Direction Problems

Every direction test question describes a walk. The person starts at a point, moves some distance in a direction, turns, and moves again. You are then asked how far the person is from the start, or how far they walked in all.

Distance is the total length of the path actually walked. It only adds up. It never reduces, and it has no direction. If you walk 5 m North and then 5 m South, the distance is 10 m.

Displacement is the straight-line gap from the starting point to the final point. It depends only on where you start and end, not on the route. In the same walk of 5 m North and 5 m South, you are back at the start, so the displacement is 0.

To find displacement, treat the walk on a grid. North and South movements cancel each other. East and West movements cancel each other. After cancelling, you are left with a net vertical gap and a net horizontal gap. These two form the sides of a right-angled triangle, and the displacement is its hypotenuse.

In exam wording, 'how far is he from the starting point' and 'shortest distance from the starting point' both mean displacement. 'Total distance travelled' or 'how far did he walk' means distance.

Key formulas to remember

Total distance: Distance = sum of all leg lengths; Add every leg. Never subtract. Direction does not matter.
Net North-South gap: NS = (total North) − (total South); A positive result means the person is North of the start. A negative result means South.
Net East-West gap: EW = (total East) − (total West); A positive result means East of the start. A negative result means West.
Displacement: Displacement = √(NS² + EW²); Valid because North-South and East-West are at right angles. Use the absolute values of NS and EW.
Straight-line case: If EW = 0 or NS = 0, displacement = |NS| or |EW|; No Pythagoras needed when the person ends on the same line as the start.
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples); Spot these to get the answer without squaring.

How to solve Distance and Displacement in Direction Problems questions

This method works for any distance or displacement question in direction tests.

1Read the question and note what is asked: total distance or distance from the start.
2Mark the start point and draw North at the top of your rough sheet.
3Draw each leg in order, with its direction and length. Turn left or right only relative to the current heading.
4For total distance, add the lengths of all legs.
5For displacement, add all North legs and subtract South legs to get NS. Do the same for East and West to get EW.
6If both NS and EW are non-zero, compute √(NS² + EW²). If one is zero, the answer is the other value.
7If the question asks for direction as well, say which side the end point lies from the start, using the signs of NS and EW.
8Match your value to the options and check that displacement is not more than distance.

Quickest way: Net-gap shortcut with triple spotting

When to use it: Use it for every MCQ on this topic, especially when the options are whole numbers.

Do not draw a neat diagram. Write only two running totals: vertical (N positive, S negative) and horizontal (E positive, W negative).
Add each leg to the right total as you read the question.
Look at the two totals. If they match a Pythagorean triple such as 3 and 4 or 5 and 12, write the hypotenuse straight away.
If one total is zero, the answer is the other total with no calculation.
Use a sanity check to eliminate options: displacement is always less than or equal to total distance.
If a question asks for distance and an option equals the sum of all legs, check that first.

Common mistakes in Distance and Displacement in Direction Problems

Subtracting legs while calculating total distance.
Students mix up distance with displacement and cancel opposite legs.
Fix: Remember that distance is the path length. Add every leg. Cancel legs only for displacement.
Adding North and South legs together to get the net gap.
Students forget that opposite directions cancel.
Fix: Always use N minus S and E minus W. Keep a signed running total.
Using Pythagoras on the sum of legs instead of the net gaps.
Students rush and treat two legs as the two sides without cancelling.
Fix: Finish the cancelling first. Apply Pythagoras only on the final NS and EW values.
Turning the wrong way after a left or right turn.
Left and right depend on the current heading, not on the page.
Fix: Write the new compass direction after every turn. Facing North, left is West. Facing East, left is North.
Giving the distance when the question asks for the shortest distance.
The word 'distance' appears in both, so students stop reading.
Fix: Treat 'shortest distance' and 'how far from the starting point' as displacement.
Leaving out the final answer unit or picking a close-looking option.
Speed under negative marking leads to careless matching.
Fix: Recheck the totals once and confirm that the value is a valid square root of NS² + EW².

Worked examples

Example 1

Ravi walks 8 m North, then turns right and walks 6 m. He then turns right and walks 8 m. How far is he from his starting point? Options: (A) 6 m (B) 8 m (C) 10 m (D) 14 m

Show the solution

Start at the origin. Walk 8 m North.
Turn right from North, so he faces East. Walk 6 m East.
Turn right from East, so he faces South. Walk 8 m South.
NS = 8 − 8 = 0. EW = 6 East.
Since NS = 0, the displacement is just 6 m.

Answer: (A) 6 m

Example 2

A man walks 5 km East, then 12 km North, then 5 km West. What is the total distance he walked, and how far is he from the start? Options: (A) 22 km and 12 km (B) 22 km and 13 km (C) 12 km and 22 km (D) 17 km and 12 km

Show the solution

Total distance = 5 + 12 + 5 = 22 km.
EW = 5 East − 5 West = 0.
NS = 12 North.
Displacement = 12 km, since EW is zero.
So distance is 22 km and displacement is 12 km.

Answer: (A) 22 km and 12 km

Example 3

Meena goes 10 m South, then 6 m West, then 6 m North, then 2 m East, and then 4 m North. How far is she from her starting point? Options: (A) 2 m (B) 4 m (C) 5 m (D) 8 m

Show the solution

List the legs: 10 m South, 6 m West, 6 m North, 2 m East, 4 m North.
Total North = 6 + 4 = 10 m. Total South = 10 m. NS = 10 − 10 = 0.
Total East = 2 m. Total West = 6 m. EW = 2 − 6 = −4, so 4 m West.
NS is zero, so displacement = 4 m.

Answer: (B) 4 m

Exam tips

Read the last line of the question first. It tells you whether you need distance, displacement or direction.
Keep a two-line running total for vertical and horizontal movement instead of drawing a full diagram.
Learn the 3-4-5 and 5-12-13 triples. Many exam questions use them so the answer is a whole number.
If an option is larger than the total distance walked, it cannot be the displacement. Eliminate it.
Questions are quick, so skip only if the turn instructions are unclear. A wrong answer costs 0.25 marks.

Practice questions from Direction Tests

Distance and Displacement in Direction Problems: frequently asked questions

What is the difference between distance and displacement in direction tests?

Distance is the total path length walked, and you add every leg. Displacement is the straight-line gap between the start and end points. Displacement is never more than the distance.

How do I find the shortest distance in direction sense questions?

Find the net North-South gap and the net East-West gap by cancelling opposite legs. Then apply √(NS² + EW²). If one gap is zero, the shortest distance is simply the other gap.

Can displacement be zero?

Yes. If the person returns to the starting point, the displacement is zero, even though the distance walked is not zero.

Do I always need Pythagoras in these questions?

No. You need it only when the person ends up away from the start both vertically and horizontally. If the person ends on the same line as the start, the answer is just the net gap on that line.