Stefan–Boltzmann law calculator

Worked example

Find L, the luminosity. Given: radius of the star r = 6.96 × 10⁸ m; surface temperature T = 5800 K; Stefan–Boltzmann constant σ = 5.67 × 10⁻⁸ W/(m² K⁴).

1. Write the equation
   L = 4πr² σ T⁴
2. Constants used
   σ = 5.67 × 10⁻⁸ W/(m² K⁴)
3. Substitute the values
   L = 4π × (6.96 × 10⁸)² × (5.67 × 10⁻⁸) × 5800⁴
4. Calculate
   L = 3.90592 × 10²⁶ W
5. Give the answer to 3 significant figures
   L = 3.91 × 10²⁶ W

Symbols and units

Stefan–Boltzmann law: rearranged

• To find luminosity: L = 4πr² σ T⁴
• To find radius of the star: r = √(L / (4πσ T⁴))
• To find surface temperature: T = (L / (4πr² σ))^{1 / 4}

Exam tip

T is the absolute temperature in kelvin. Raise only T to the fourth power.

Common questions

What is the formula for stefan–boltzmann law?

The equation is L = 4πr² σ T⁴, where L is the luminosity (W); r is the radius of the star (m); T is the surface temperature (K); σ is the Stefan–Boltzmann constant (W/(m² K⁴)).

What units should I use for stefan–boltzmann law?

In SI units: luminosity in W, radius of the star in m, Stefan–Boltzmann constant in W/(m² K⁴), surface temperature in K. The calculator also accepts common alternatives such as kilometres, grams or minutes, and converts them for you.

How do I rearrange L = 4πr² σ T⁴?

Rearranging gives these forms: to find luminosity: L = 4πr² σ T⁴. to find radius of the star: r = √(L / (4πσ T⁴)). to find surface temperature: T = (L / (4πr² σ))1 / 4.