Length Contraction Calculator Formula

Understand the math behind the length contraction calculator. Each variable explained with a worked example.

Use the Length Contraction Calculator

Formulas Used

Contracted Length

contracted_length = proper_length * sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2))

Length Reduction (%)

contraction_pct = (1 - sqrt(1 - pow(velocity, 2) / pow(2.998e8, 2))) * 100

Variables

VariableDescriptionDefault
proper_lengthProper Length (L0)(m)100
velocityVelocity(m/s)269800000

How It Works

Lorentz Length Contraction

Objects moving at relativistic speeds appear shorter in the direction of motion.

Formula

L = L0 * sqrt(1 - v^2/c^2) = L0 / gamma

  • *L0* = proper length (measured at rest)
  • *v* = relative velocity

    • Contraction occurs only along the direction of motion. Transverse dimensions are unaffected.

    Worked Example

    A 100 m spaceship at 0.9c.

    proper_length = 100velocity = 269800000
    1. 01beta = 0.9, gamma = 2.294
    2. 02L = L0 / gamma = 100 / 2.294
    3. 03L = 43.59 m
    4. 04The ship appears 56.4% shorter.

    Frequently Asked Questions

    Would a moving object look visually squished?

    Not exactly. Due to the finite speed of light, a moving object appears rotated (Terrell rotation) rather than simply compressed. The contraction is real but the visual appearance is more complex.

    Does the object actually shrink?

    In the moving object's own frame, it is not contracted at all. Length contraction is a real physical effect of how different frames measure space, not an illusion.

    At what speed is the contraction noticeable?

    At 10% of c, contraction is only 0.5%. It becomes dramatic above 80% of c, where the length drops below 60% of the rest length.

    Ready to run the numbers?

    Open Length Contraction Calculator