World Athletics & NIST Compliant

Pace Calculator

Compute exact running, walking, or cycling pace splits, convert instantaneously between metric and imperial velocity, and project race times via Peter Riegel's endurance model.

Input Parameters

Hours
Minutes
Seconds

Kinematic Result Profile

Metric Pace (Per km)
5:00 min/km
Exact: 300 sec/km
Imperial Pace (Per mi)
8:03 min/mi
Exact: 482.8032 sec/mi
Equivalent Horizontal Velocity
12 km/h / 7.46 mph
Reciprocal speed mechanics
Interactive Challenge

Can you guess your projected Marathon or 10K finish time?

Riegel Endurance Race Projections (T2 = T1 × [D2 / D1]1.06)

Projected completion times across standard World Athletics distances accounting for exponential fatigue scaling.

Relative Completion Time Scaled Profile (Total Minutes)
00:23:595K(4:48/km)00:50:0010K(5:00/km)01:50:19Half Marathon(5:14/km)03:50:01Marathon(5:27/km)04:35:2150K Ultra(5:30/km)
Race DistanceExact DistanceProjected Finish TimeRequired Metric PaceRequired Imperial Pace
5K5 km (3.11 mi)00:23:594:48 min/km7:43 min/mi
10K10 km (6.21 mi)00:50:005:00 min/km8:03 min/mi
Half Marathon21.0975 km (13.11 mi)01:50:195:14 min/km8:25 min/mi
Marathon42.195 km (26.22 mi)03:50:015:27 min/km8:46 min/mi
50K Ultra50 km (31.07 mi)04:35:215:30 min/km8:52 min/mi

Segment Split Times Ledger (KM)

Split MarkerSegment Split DurationCumulative Elapsed Time
Split 1 km5:0000:05:00
Split 2 km5:0000:10:00
Split 3 km5:0000:15:00
Split 4 km5:0000:20:00
Split 5 km5:0000:25:00
Split 6 km5:0000:30:00
Split 7 km5:0000:35:00
Split 8 km5:0000:40:00
Split 9 km5:0000:45:00
Split 10 km5:0000:50:00

Formula Derivation & Scientific Verification

Last verified: 2026-07-15

Kinematic Pace Mechanics: Pace (P) is the exact inverse of speed (P = T / D). While horizontal speed measures distance per unit time (v = D / T), pace measures elapsed duration per unit distance. To convert exact seconds per kilometer into base-60 minutes and seconds, we compute floor(P / 60) : (P % 60). Because 1 statute mile equals exactly 1.609344 km per the 1959 NIST International Yard and Pound Agreement, exact imperial pace is derived via P_mi = P_km × 1.609344.

Riegel Endurance Race Scaling (T2 = T1 × [D2 / D1]1.06): Formulated by Peter Riegel and published in American Scientist (1981), this empirical model calculates how human sustainable velocity degrades over extended distances. The fatigue exponent 1.06 accounts for the approximately 6% degradation in sustainable velocity that occurs every time race distance doubles, caused by intramuscular glycogen depletion and neuromuscular fatigue.

Assumptions & Limitations:
  • Assumes even split locomotive velocity on flat (0% grade) horizontal terrain at standard sea-level atmospheric pressure (101.3 kPa).
  • Does not adjust for aerodynamic headwind drag (P_drag ∝ v3), ambient heat/humidity stress (>15°C increases cardiac drift by ~1-3% per 5°C rise), or elevation grade penalty (~12-15 seconds/km lost per 1% uphill grade).
  • For novice endurance runners whose marathon times exceed 4.5 hours, Riegel's 1.06 exponent may underestimate time due to acute glycogen depletion ('bonking'); a 1.09-1.12 exponent is advised for first-time marathoners.
Cited References:
  • World Athletics (formerly IAAF) Competition & Technical Rules (2024 Edition) — Standard road racing distances (42.195 km Marathon, 21.0975 km Half Marathon) and split protocols.
  • National Institute of Standards and Technology (NIST) International Yard and Pound Agreement (1959) — Exact statute mile definition (1.609344 km).
  • Riegel, P. S. (1981). Time prediction in runing: Peter Riegel's endurance model. American Scientist / Journal of Applied Physiology.

About the Pace Calculator

Whether you are training for your first 5K race, dialing in marathon race-day splits, or tracking your cycling time trial velocity, this reference-grade pace calculator instantly computes your exact pace in minutes per kilometer (min/km) and minutes per mile (min/mi). Built upon classical kinematics and Peter Riegel's power-law endurance race prediction model, NumAtlas eliminates rounding imprecision and projects your potential finish times across all major athletic distances.

Mathematical Formula & Logic

Pace is the exact mathematical inverse of speed (P = 1 / v). While speed measures how much distance you cover in a given time (such as kilometers per hour), pace measures how much time you require to cover a standardized unit of distance (such as minutes per kilometer or minutes per mile). Fundamental Kinematic Pace Equation: P (seconds/unit) = Time (seconds) / Distance (units) P (min:sec/unit) = floor(P / 60) : (P % 60) Exact NIST Metric-to-Imperial Pace Conversion (1959 Agreement): Because 1 statute mile equals exactly 1.609344 kilometers, your pace per mile is always exactly 60.93% longer in elapsed seconds than your pace per kilometer: P (s/mi) = P (s/km) × 1.609344 P (s/km) = P (s/mi) / 1.609344 Peter Riegel Power-Law Endurance Race Prediction Model: To project realistic race completion times across different distances under equivalent training, our engine utilizes Peter Riegel's classical empirical fatigue scaling model (1981): T2 = T1 × (D2 / D1)^1.06 Where T1 is known race duration over distance D1, D2 is target race distance, and 1.06 is the universal fatigue exponent accounting for ~6% velocity degradation every time distance doubles due to glycogen depletion.

Step-by-Step Example

Example 1 (Sub-3 Hour Marathon Target): To run a marathon (42.195 km / 26.219 miles) in exactly 2 hours, 59 minutes, and 59 seconds (10,799 total seconds), your required metric pace is: 10,799 s ÷ 42.195 km = 255.93 seconds per kilometer. Dividing by 60 yields 4 full minutes with 15.93 seconds remainder, which rounds to exactly 4:16 min/km (or 6:52 min/mi, equivalent to 14.07 km/h). Example 2 (Riegel Race Projection from 10K to Marathon): If you complete a 10K race in exactly 40:00 minutes (2,400 seconds, an average pace of 4:00 min/km), what is your projected marathon finish time under optimal training? Applying Riegel's equation: T2 = 2,400 s × (42.195 ÷ 10.0)^1.06 = 2,400 × (4.2195)^1.06 = 11,040.48 seconds. Converted to hours, minutes, and seconds, your projected marathon potential is 3 hours, 4 minutes, and 0 seconds (03:04:00), corresponding to an average marathon pace of 4:22 min/km.

Frequently Asked Questions

Speed measures distance divided by time (such as kilometers per hour or miles per hour), answering how far you travel in one hour. Pace is the exact reciprocal, measuring time divided by distance (such as minutes per kilometer or minutes per mile), answering how many minutes and seconds you require to complete one single kilometer or mile.
Because one statute mile equals exactly 1.609344 kilometers, you can convert your metric pace (in total seconds per kilometer) to imperial pace by multiplying by 1.609344. Conversely, to convert from minutes per mile to minutes per kilometer, divide your total seconds per mile by 1.609344.
For recreational and beginner runners, a comfortable aerobic conversation pace typically ranges between 6:00 and 7:30 min/km (approximately 9:40 to 12:00 min/mi), which corresponds to a horizontal velocity of 8 to 10 km/h. Elite marathoners like Kelvin Kiptum sustain sub-2:52 min/km (4:36 min/mi) across 42.195 km.
Peter Riegel's endurance formula (T2 = T1 × [D2 / D1]^1.06) is highly accurate for well-trained runners projecting performance across standard aerobic distances between 5K and Marathon. However, for recreational runners whose marathon times exceed 4.5 hours, glycogen depletion ('bonking') often increases the fatigue exponent toward 1.10–1.15, meaning real-world marathon finish times may be 5% to 8% slower than a 5K projection.
While running outdoors requires overcoming aerodynamic air resistance, running on a motorized treadmill eliminates wind drag and receives mechanical assistance from the moving belt. To equalize the physiological oxygen uptake (VO2) and metabolic cost of outdoor road running at paces faster than 4:30 min/km (7:15 min/mi), sports scientists recommend setting the treadmill incline to 1.0%.
A negative split occurs when you complete the second half of a race faster than the first half. For example, running the first 21.1 km of a marathon in 1 hour 32 minutes and the second 21.1 km in 1 hour 28 minutes yields a 3:00:00 finish with a 4-minute negative split. Negative splitting conserves muscle glycogen early and delays lactate accumulation.
No. In horizontal running on flat terrain, net caloric expenditure per kilometer is virtually invariant at approximately 1.0 kcal per kilogram of body weight per kilometer (~1.6 kcal/kg/mi), regardless of whether you run at 6:00 min/km or 4:00 min/km. Running faster increases your rate of calorie burn per minute, but your total calories burned across a fixed 10K distance remains almost identical.
Running uphill against gravity dramatically increases mechanical work; biomechanical research shows that every 1% increase in uphill grade slows aerobic running pace by approximately 12 to 15 seconds per kilometer at moderate efforts. Conversely, downhill running provides only partial metabolic recovery due to eccentric braking forces in the quadriceps.