โฑ๏ธRelated Rates Calculator
Pick a classic scenario and get the unknown rate with the full implicit-differentiation derivation.
Free Related Rates Calculator
Related rates problems are notorious for being more about setup than calculus โ our free related rates calculator handles both. Pick a classic scenario like a sliding ladder, an expanding balloon, or a draining cone, enter the known values and rates, and get the unknown rate instantly with the full derivation shown.
Each scenario starts from its governing geometric equation, which is then differentiated implicitly with respect to time, with every chain-rule term explained before the known values are substituted in and solved. Sign conventions (rates increasing vs. decreasing) are stated explicitly to avoid the most common source of related-rates mistakes.
Pick your scenario to get started. No sign-up, no downloads, no limits.
About the Related Rates Calculator
Features
Five scenarios
Sliding ladder, expanding sphere, draining cone, growing shadow, and expanding circle.
Full derivation
Governing equation, implicit differentiation, and substitution shown for every scenario.
Labelled diagrams
A simple diagram for each scenario matching the input variables.
Sign conventions
Explicit statement of whether each rate is increasing or decreasing.
Frequently Asked Questions
How does the related rates calculator work?
Pick a scenario, enter the known values and rates, and it derives the governing equation, differentiates it with respect to time, and solves for the unknown rate.
Is it free?
Completely free, no sign-up, downloads, or limits.
What scenarios are supported?
Sliding ladder, expanding sphere/balloon, draining/filling cone, growing shadow, and expanding circle, each with its own setup.
Does it show the full derivation?
Yes, from the governing equation through implicit differentiation to the final substituted answer.
Does it show a diagram?
Yes, each scenario includes a labelled diagram matching the input variables.
How are sign conventions handled?
The calculator states explicitly whether a rate is positive (increasing) or negative (decreasing) for each scenario.
What if my inputs are physically impossible?
E.g. a distance exceeding the ladder length is flagged with a clear validation message.
Do I need to convert units myself?
Units are carried through as labels you provide; no conversion is performed or required.
Is my input stored anywhere?
No; everything runs in your browser and isn't stored or sent anywhere.
Does it work on mobile?
Yes, fully responsive, including the diagrams and step-by-step panel.