Circular Track: Formulas, Rules and Solved Questions

Circular Track questions use circumference, speed, time and relative speed to solve problems on circular paths. The main cases involve runners moving in the same or opposite directions, meeting at intervals, overtaking, completing laps and comparing race results. The correct relative speed and distance around the track determine the solution.

On this page

What Is a Circular Track?

A circular track is a closed path whose total distance is its circumference, calculated as 2πr or πd. In track problems, a runner’s distance is expressed in terms of complete laps or fractions of the circumference.

If the track has radius r, its circumference is C = 2πr. If its diameter is d, C = πd. A runner moving at speed v completes one lap in time C/v. For n laps, the distance is nC and the time is nC/v. Use consistent units for distance and speed before calculating.

Circular Track Formula & Tricks

Important Formulas

Track circumference
C = 2πr = πd

Here, r is the radius, d is the diameter and C is the total length of the circular track.

Time for one lap
t = C/v

A person moving with speed v takes circumference divided by speed to complete one lap.

Same-direction relative speed
v_rel = |v₁ − v₂|

For runners moving in the same direction, the faster runner gains on the slower runner at the difference of their speeds.

Opposite-direction relative speed
v_rel = v₁ + v₂

For runners moving towards each other, their relative speed is the sum of their speeds.

First meeting from the same point
t = C/(v₁ + v₂) for opposite directions; t = C/|v₁ − v₂| for the same direction

When both runners start together from one point, one full circumference is the relative distance needed for the first meeting or catch-up.

Number of meetings
n = (v₁ + v₂)t/C for opposite directions; n = |v₁ − v₂|t/C for the same direction

The value n gives the number of complete relative circumferences covered during time t.

Quick Tricks

Use one circumference as the relative distance

When two runners start together from the same point, meeting or overtaking occurs whenever their relative distance becomes one complete circumference.

Example: On a 400 m track, speeds are 8 m/s and 6 m/s in the same direction. First overtaking time = 400/(8 − 6) = 200 seconds.
Choose sum or difference before calculating

Use the sum of speeds for opposite directions and the difference of speeds for the same direction. This prevents the most common relative-speed error.

Example: For speeds 10 m/s and 4 m/s, relative speed is 14 m/s in opposite directions but 6 m/s in the same direction.
Convert speed units first

Use 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h. Distance and speed must use compatible units.

Example: A speed of 54 km/h equals 54 × 5/18 = 15 m/s.

Circular Track Concepts

Laps, Distance and Time

For n laps on a circular track, distance equals n times the circumference, so time equals nC/v.

If the radius is r, C = 2πr. Therefore, distance for n laps is 2πrn and time at speed v is 2πrn/v. For a fraction of a lap, multiply the circumference by that fraction.

Example: A runner completes 3 laps of a 70 m circumference track at 7 m/s. Distance = 3 × 70 = 210 m, and time = 210/7 = 30 seconds.

Meeting in Opposite Directions

When two runners move in opposite directions from the same point, their first meeting time is C/(v₁ + v₂).

Their distances add because both runners move towards each other along the track. After every additional time interval of C/(v₁ + v₂), they meet again. In time T, the number of meetings is (v₁ + v₂)T/C when the result represents complete meeting intervals.

Example: On a 600 m track, two runners move in opposite directions at 8 m/s and 7 m/s. First meeting time = 600/(8 + 7) = 40 seconds.

Overtaking in the Same Direction

When two runners move in the same direction, the faster runner overtakes the slower runner after covering one relative circumference at relative speed |v₁ − v₂|.

For runners starting together, first overtaking time is C/|v₁ − v₂|. If they start with an initial gap s measured along the direction of the faster runner, use s/|v₁ − v₂| for the first catch-up when no extra full lap is required. Each later overtaking adds one circumference to the relative distance.

Example: On a 500 m track, speeds are 9 m/s and 5 m/s. First overtaking time = 500/(9 − 5) = 125 seconds.

Different Starting Points

If runners start at different points, use the initial arc gap along the direction in which the faster runner must close it, together with the appropriate relative speed.

For the same direction, catch-up time is gap/(faster speed − slower speed). For opposite directions, meeting time is gap/(sum of speeds), where the gap is the shorter or specified arc distance between them along their paths. If the runners start from the same point, the required relative distance becomes one circumference.

Example: Two runners are 120 m apart on a track and move towards each other at 8 m/s and 7 m/s. Meeting time = 120/(8 + 7) = 8 seconds.

Race Results and Lead

In a race of track length L, a runner’s finishing time is L/v, and the distance covered by another runner in that time is speed multiplied by the winner’s finishing time.

If A finishes a one-lap race in time t and B runs at speed v_B, B covers v_Bt. A’s lead over B is L − v_Bt. If A beats B by a distance x, then x = L(1 − v_B/v_A), assuming both start together and A is faster.

Example: In a 400 m race, A runs at 8 m/s and B at 6 m/s. A finishes in 400/8 = 50 seconds. B covers 6 × 50 = 300 m, so A wins by 100 m.

Circular Track Video Lessons

Watch short topic-wise lessons for quick revision.

11 Lessons
Lesson 1 of 11 Quick Revision

Circular Track Meeting and Lapping

Learn how to solve circular track problems involving two or more runners meeting, overtaking, and completing laps using relative speed and track length.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Circular Track Lessons Scroll to explore →

Practice Circular Track Questions

Practise published questions related to this topic.

1Successive discounts of 16% and 22% are equivalent to what single discount?→ 2Which is the most significant 4-digit number exactly divisible by 88?→ 3A person sells an article at the loss of 10 percent. Cost price and selling price both are decreased by 20 percent. What will be the new loss percentage?→ 4A shopkeeper marks an article 30% above its cost price. He then allows a discount of 20% on the marked price and still makes a profit of ₹32. Find the cost price of the article.→ 5The square root of the sum of squares of two numbers is 25. If one number is 15, find the other number.→ 6If √x = 5, what is the value of √(x+25)?→ 7The marked price of an article is 60 percent more than its cost price. What maximum discount percentage can be offered on this article so that there is no loss?→ 8Find the least number that must be added to 5600 to make it a perfect square.→ 9Find the single discount equivalent to two successive discounts of 29% and 28%.→ 10The product of two numbers is 6760 and their H.C.F. is 13. How many such pairs can be formed?→

Circular Track Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Circular Track Revision Points

Use these formulas and rules for quick calculation of circular track questions.

  • Circumference C = 2πr = πd.
  • Distance for n laps = nC.
  • Time = distance/speed.
  • Same-direction relative speed = absolute difference of speeds.
  • Opposite-direction relative speed = sum of speeds.
  • For the same starting point, first opposite-direction meeting time = C/(v₁ + v₂).
  • For the same starting point, first same-direction overtaking time = C/|v₁ − v₂|.
  • For a known initial gap, divide the gap by relative speed instead of using the full circumference automatically.

Circular Track FAQs

What is the formula for the circumference of a circular track?

The circumference is C = 2πr, where r is the radius. Since d = 2r, it can also be written as C = πd.

What is the relative speed on a circular track when two people move in the same direction?

Relative speed is |v₁ − v₂|. For speeds 12 m/s and 8 m/s, it is 4 m/s.

What is the relative speed when two runners move in opposite directions?

Relative speed is v₁ + v₂. For speeds 10 m/s and 6 m/s, it is 16 m/s.

Two runners move in opposite directions on a 300 m track at 5 m/s and 10 m/s. When will they first meet?

Their relative speed is 5 + 10 = 15 m/s. First meeting time = 300/15 = 20 seconds.

Two runners move in the same direction on a 400 m track at 9 m/s and 7 m/s. When will the faster runner first overtake the slower runner?

Relative speed = 9 − 7 = 2 m/s. Overtaking time = 400/2 = 200 seconds.

How is the time calculated when runners start from different points?

Time equals the initial arc gap divided by relative speed. Use the speed difference for the same direction and the speed sum for opposite directions.

Continue learning Circular Track on PrepShots

Continue on PrepShots