Free · Physics models
Linearized Isotropic Thermal Expansion
Relate signed length and volume changes to temperature change and entered constant expansion coefficients.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
For α = 10⁻⁵ K⁻¹ and a 100 K increase, 2 m length grows by 2 mm and 1 m³ grows by 0.003 m³.
ΔL = α L0 ΔT; ΔV = β V0 ΔT; β ≈ 3α for isotropic small expansion.
Worked example
Enter these known values and leave the other values blank.
- Initial temperature
- 20 °C
- Final temperature
- 120 °C
- Linear expansion coefficient
- .00001 / K
- Initial length
- 2 m
- Initial volume
- 1 m³
- Final length
- 2.002 m
- Change in length
- 0.002 m
- Volumetric expansion coefficient
- 0.00003 / K
- Final volume
- 1.003 m³
- Change in volume
- 0.003 m³
Assumptions and limitations
- Assume a freely expanding, isotropic material with a constant linear coefficient over the entered temperature interval. Constraints and thermal stress are omitted.
- The β = 3α relationship is a first-order small-strain approximation. It does not use the exact cubic volume ratio (1+αΔT)³.
- Cooling gives negative changes for positive α; materials with negative α can contract on heating. Coefficients are entered per temperature interval, without absolute-temperature offsets.
- Initial and final lengths and volumes must stay positive. Temperatures cannot be below absolute zero.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Can the change be negative?
Yes. Cooling a positive-expansion material produces negative length and volume changes. Negative-expansion materials reverse that sign relation.
Is beta exactly three alpha?
The relation is a first-order isotropic small-strain approximation. Finite cubic changes and changing coefficients require a fuller model.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.