Home

Related Rates Calculator

Related-rates solver. Choose a sphere, circle or square; given one quantity and its rate, find the related rate—with the relation equation.

Show more ▾
Differentiate the relationship with respect to t to find the unknown rate
Geometric related rates
Instantaneous value
Rate (units/s)

📖 Tutorial｜Related Rates

1. Definition
Related rates: two quantities x(t),y(t) are related; differentiate w.r.t. t to find one rate from the other.
2. Symbols
SymbolMeaning
r, sInstantaneous value
dr/dt, ds/dtKnown rate
dV/dt, dA/dtWanted rate
3. How it works
  • We differentiate the geometric relation w.r.t. t;
  • Plug in the instantaneous value and known rate;
  • Solve for the wanted rate.
4. Steps
  1. Choose a model;
  2. Enter the instantaneous value;
  3. Enter the known rate and click Calculate.
5. Example
Example: Balloon r=5, dr/dt=2. dV/dt=4πr²·dr/dt=200π≈628.3.
6. Pitfalls
Positive/negative rate means increasing/decreasing;
Plug in instantaneous values, not functions;
Set up the relation correctly.

❓ FAQ｜Related Rates

How to start a related-rates problem?
Draw, label, write the geometric relation, differentiate w.r.t. t, plug in known values.
Why differentiate w.r.t. t?
Every quantity changes with time; the chain rule turns geometry into rates.
What does a negative rate mean?
The quantity is decreasing, e.g. dr/dt=−2 means shrinking at 2 cm/s.
Sphere volume rate formula?
dV/dt=4πr²·dr/dt from V=4/3πr³.
Common related-rates scenarios?
Inflating balloons, sliding ladders, moving shadows, water pouring; all hinge on differentiating w.r.t. t.