Related Rates Calculator
Calculate how fast the area or volume of a circle, sphere, or cylinder changes given the rate at which its radius is changing. Useful for calculus related-rates problems in physics and engineering.
About this calculator
Related rates problems apply implicit differentiation with respect to time. For a circle, area A = πr², so dA/dt = 2πr · (dr/dt). For a sphere, volume V = (4/3)πr³, giving dV/dt = 4πr² · (dr/dt). For a cylinder of fixed height h, lateral surface area S = 2πrh, so dS/dt = 2πrh · (dr/dt). In each case you multiply the geometric derivative by the rate of change of the radius to obtain the rate of change of the quantity of interest. Units follow naturally: if r is in meters and dr/dt in m/s, the output is in m²/s or m³/s.
How to use
A spherical balloon is being inflated so its radius grows at dr/dt = 0.5 cm/s. At the moment the radius is r = 4 cm, how fast is the volume increasing? Select shape = sphere, enter radius = 4, radius_rate = 0.5. The formula gives: dV/dt = 4π·(4²)·(0.5) = 4π·16·0.5 = 32π ≈ 100.53 cm³/s. This tells you the balloon's volume is growing at about 100.5 cubic centimetres per second at that instant.
Frequently asked questions
What are related rates in calculus and when do you use them?
Related rates is a technique that uses the chain rule to relate the rates of change of two or more quantities that are linked by a geometric or physical equation. You use it whenever one measurable quantity changes over time and you need to find how fast a related quantity changes at a specific instant. Common applications include expanding ripples on water, inflating balloons, sliding ladders, and filling tanks.
How do you set up a related rates problem for a sphere?
Start by writing the volume formula V = (4/3)πr³. Differentiate both sides with respect to time t using the chain rule: dV/dt = 4πr² · dr/dt. Identify which quantities are known at the instant of interest — typically the current radius and the rate dr/dt — and substitute them in. The result is the instantaneous rate of volume change at that moment.
Why does the rate of area or volume change depend on the current radius?
Because area and volume are nonlinear functions of radius, their rates of change are not constant — they grow as the shape gets larger. Mathematically, the derivative dA/dt = 2πr·(dr/dt) shows that even with the same dr/dt, a larger r produces a faster area increase. This is why a balloon seems to expand slowly at first and then very rapidly once it is already large.