# Newton’s Law of Cooling Calculator

URL: https://speedycalculator.com/calculators/physics/newtons-law-of-cooling/
Last updated: 2026-10-25

> This calculator predicts the temperature of a cooling object over time using Newton’s law of cooling, T(t) = Tₐ + (T₀ − Tₐ)e^(−kt). Use it for coffee, food safety or forensic estimates.

## Inputs

- Initial temperature (°C, °F, K)
- Surrounding temperature (°C, °F, K)
- Cooling constant k (per minute)
- Time (min, h, s)

## Results

- Temperature after time
- Time to halve the difference

## Formula

```
T(t) = T_ambient + (T₀ − T_ambient) × e^(−k t)
```

## Explanation

An object cools fastest when it is much hotter than its surroundings, and slows as it approaches room temperature.

### How to use the Newton’s Law of Cooling Calculator

- **Initial temperature**
- **Surrounding temperature**
- **Cooling constant k**
- **Time**
- Want to work backwards? Type a value into **Temperature after time** and **Time** is calculated for you.

Results update as you type, and every field accepts the units shown in its menu.

### How it works

The temperature difference decays exponentially with a rate constant k that depends on the object.

### Inputs explained

FieldWhat to enterInitial temperatureUnits: °C, °F, KSurrounding temperatureUnits: °C, °F, KCooling constant kIn per minuteTimeUnits: min, h, s

### What each result means

- **Temperature after time** = [Surrounding temperature] + ([Initial temperature] - [Surrounding temperature])×exp(-[Cooling constant k]×[Time])
- **Time to halve the difference** = ln(2)÷[Cooling constant k]

### Example values

How the main result changes when **Time** changes and everything else stays at the example values above:

Time (min)Temperature after time (°C)1063.21554.120472541.53037.24031.2

## Worked example

Inputs — Initial temperature: 90 °C; Surrounding temperature: 22 °C; Cooling constant k: 0.05 per minute; Time: 20 min.

Results — **Temperature after time = 47 °C**; Time to halve the difference = 13.9 min.

## FAQ

**How do I find k?**
Measure the temperature at two times and solve the equation for k.

**Is the law exact?**
It is a good approximation when the temperature difference is moderate.

**What formula does the Newton’s Law of Cooling Calculator use?**
T(t) = T_ambient + (T₀ − T_ambient) × e^(−k t).

**Can I work backwards from the result?**
Yes. Type the result you want into Temperature after time and the calculator finds the matching time.