Calculateus

Stefan-Boltzmann Law Calculator

Calculate the total power radiated by a blackbody based on its temperature and surface area.

Result

Radiated Power
63,200,695.06 W

Formula: P = εσAT⁴, using the Stefan-Boltzmann constant σ ≈ 5.670 × 10⁻⁸ W/(m²·K⁴). Because power scales with the fourth power of temperature, small temperature increases produce large increases in radiated power - doubling absolute temperature increases radiated power sixteenfold.

About the Stefan-Boltzmann Law

The Stefan-Boltzmann Law Calculator finds the total power a hot object radiates as electromagnetic energy, based on its temperature, surface area, and emissivity. It's used across astronomy and thermal engineering, from estimating a star's radiated power to checking how much heat a hot surface loses to radiation.

How It Works

You enter a temperature in Kelvin, a surface area in square meters, and an emissivity value between 0 and 1, where 1 represents a perfect blackbody and lower values represent real reflective or polished materials. The calculator raises the temperature to the fourth power and multiplies it by the Stefan-Boltzmann constant, the area, and the emissivity to get total radiated power in watts.

P = εσAT⁴, where σ, the Stefan-Boltzmann constant, is about 5.670374 × 10⁻⁸ W/(m²·K⁴), T is absolute temperature in Kelvin, A is surface area in m², and ε is emissivity.

Formula & Methodology

For a quick hand estimate without recomputing the full formula, compare two temperatures as a ratio: because power depends on T⁴, doubling an object's absolute temperature multiplies its radiated power by 2⁴, or sixteen times, regardless of its size or emissivity. This ratio shortcut is often faster than working through the full εσAT⁴ calculation when only a relative comparison between two temperatures is needed.

Examples

Sun-Like Surface Temperature

At the Sun's approximate surface temperature of 5,778 K, a 1 m² patch of blackbody surface with emissivity 1 radiates about 6.32 × 10⁷ W, roughly 63.2 million watts per square meter.

A Warm Object at Room Temperature

A 2 m² surface at 300 K with an emissivity of 0.9, typical for a matte, non-metallic material, radiates about 826.7 W, showing how much less power ordinary warm objects emit compared to a star's surface.

Advantages

  • Demonstrates just how steeply radiated power rises with temperature, since the fourth-power relationship is easy to underestimate mentally.
  • Accounts for real-world emissivity, not just the idealized blackbody case, so it applies to actual materials and surfaces.
  • Works at both astronomical and everyday scales, from stellar surfaces to a warm equipment enclosure.

Common Mistakes

  • Entering a Celsius temperature instead of Kelvin, an easy slip since many temperature-related calculators default to Celsius.
  • Assuming an emissivity of 1 for a real material, when most everyday surfaces, aside from matte black finishes, radiate well below the ideal blackbody rate.
  • Using only the projected or visible area of an object rather than its full radiating surface area, which understates the true radiated power.

Edge Cases to Watch For

  • Because power scales with the fourth power of temperature, a 10% error in the temperature input produces roughly a 46% error in the radiated power output - small mistakes get amplified dramatically.
  • The temperature must be entered in Kelvin. There's no unit-detection or conversion built in, so entering a Celsius or Fahrenheit value produces a result with no physical meaning.
  • The emissivity field is restricted to a 0-1 range in the input control, but the underlying calculation applies no separate clamp of its own, so it simply multiplies through whatever value it receives.

Common Use Cases

  • Astronomy students or hobbyists estimating a star's luminosity from its surface temperature and size.
  • Engineers estimating radiative heat loss from a hot enclosure, pipe, or component.
  • Educators demonstrating why small temperature changes cause large changes in radiated energy.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

Why does radiated power depend on the fourth power of temperature?

The T⁴ dependence emerges from integrating Planck's blackbody radiation law across all wavelengths, reflecting that hotter objects don't just radiate more at every wavelength, they also shift their peak emission toward shorter, more energetic wavelengths (per Wien's displacement law) - the combined effect compounds into this steep fourth-power relationship, which is why even modest temperature changes dramatically affect an object's radiated energy.

Related calculators

Ohm's Law CalculatorCalculate voltage, current, resistance, or power using Ohm's Law - solve for whichever value is missing.Projectile Motion CalculatorCalculate the range, maximum height, and flight time of a projectile launched at an angle.Free Fall & Kinematics CalculatorCalculate the distance fallen and final velocity of an object in free fall after a given time.Force Calculator (Newton's Second Law)Calculate force, mass, or acceleration using Newton's second law of motion (F = ma).

Conclusion

This calculator makes the Stefan-Boltzmann relationship concrete by turning temperature, area, and emissivity into an actual wattage figure. Because the fourth-power term dominates the result, getting the temperature input right in Kelvin matters more than any other variable here.