Circle and sphere
Connect radius to area using A = πr², or radius to spherical volume using V = 4πr³/3.
Related Rates Calculator with Steps
Choose a standard geometry or motion model, enter the measurements at the instant of interest, and calculate the unknown rate. The calculator keeps increasing and decreasing signs, shows the implicit-differentiation steps, and runs locally in your browser.
Choose a related-rates model
Use one consistent unit system. Enter decreasing or draining rates as negative.
Try an example
Keyboard shortcut: Ctrl/⌘ + Enter.
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Unknown rate
Expanding circleRelated equation
Choose a model and enter its values.
Result
The calculated rate will appear here.
The sign and units will be explained here.
Step-by-step related-rates solution
Choose a model and calculate to see the setup, differentiation, substitution, and result.
Calculator methodology and limits
- Developed and reviewed by
- Starlight Robotics calculator team
- Last reviewed
- Method
- Select a model → differentiate its constraint with respect to time → substitute instantaneous values → isolate the requested rate.
- Numerical limits
- Finite decimal or scientific-notation inputs with magnitudes up to 10150; geometry dimensions must satisfy the displayed model constraints.
- Units
- All inputs must already use the same selected length and time units. No hidden conversion is applied.
- Privacy
- Inputs and results stay in your browser; no solving service receives them.
How to solve a related rates problem
- Identify the quantities that change with time and the measurements known at the instant in question.
- Choose an equation that relates those quantities. For a sliding ladder, for example, the fixed length gives x² + y² = L².
- Differentiate the equation with respect to time. Apply the chain rule to every changing quantity.
- Only after differentiating, substitute the instantaneous measurements and known signed rates.
- Solve for the unknown derivative and interpret its sign with the correct units.
\[F(x(t),y(t),\ldots)=0\quad\Longrightarrow\quad F_x\frac{dx}{dt}+F_y\frac{dy}{dt}+\cdots=0\]
A positive result means the requested quantity is increasing; a negative result means it is decreasing. The magnitude alone does not communicate direction.
Supported related-rates models
Sliding ladder
Use the Pythagorean constraint x² + y² = L² for a rigid ladder against perpendicular ground and wall.
Conical and cylindrical tanks
Convert a signed volume flow rate into water-level change. The cone assumes similar cross-sections, so r/h = R/H.
Moving objects
Use s² = x² + y² for perpendicular separation components and their signed component rates.
Changing rectangle
Apply the product rule to A = ℓw when both length and width may change.
Worked examples
Expanding circle
If r = 5 cm and dr/dt = 2 cm/s, then dA/dt = 2π(5)(2) = 20π ≈ 62.8319 cm²/s.
Sliding ladder
For a 13-ft ladder with x = 5 ft and dx/dt = 0.6 ft/s, y = 12 ft and dy/dt = −(5/12)(0.6) = −0.25 ft/s.
Conical tank
For R = 10 cm, H = 30 cm, h = 4 cm, and dV/dt = 10 cm³/s, similar triangles give dh/dt = 90/(16π) ≈ 1.79049 cm/s.
Method reference: OpenStax Calculus Volume 1, Section 4.1: Related Rates.
Related rates calculator FAQs
What are related rates?
They are rates of change connected by an equation among quantities that depend on the same variable, usually time. Implicit differentiation turns that equation into a relationship among derivatives.
Should a shrinking or draining rate be negative?
Yes. Use a positive sign for increasing, filling, or moving in the positive coordinate direction. Use a negative sign for shrinking, draining, or moving in the negative direction.
Why should I substitute values after differentiating?
A measurement such as radius is an instantaneous value of a quantity that changes with time. Substituting it before differentiating can incorrectly erase its derivative.
Which units should I choose?
Choose one length unit and one time unit, then express every input in that system. Linear rates use length/time, area rates use length²/time, and volume rates use length³/time.
Does the ladder model include the ladder's own movement?
The ladder is assumed rigid, with constant length, and the wall and ground are assumed perpendicular. The variables x and y are the base distance and top height.
What does the conical-tank model assume?
The tank is an inverted right circular cone, the liquid surface remains horizontal, and the liquid forms similar cones with r/h = R/H. The entered water depth must not exceed the tank height.
Can this solve any related-rates word problem?
No. It solves the seven named models. Problems involving shadows, angles, nonstandard containers, or different constraints need equations specific to their geometry.
Does my calculation leave the browser?
No. Calculations, copied text, and downloaded steps are created locally on your device.