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Related Rates Calculator

Last updated: • Math Calculators

Circle Area: A = πr²
Square Area: A = s²
Rectangle Area: A = lw
Triangle Area: A = (1/2)bh
Sphere Volume: V = 4/3πr³
Cylinder Volume: V = πr²h
Cone Volume: V = (1/3)πr²h
Cube Volume: V = s³
Pythagorean Theorem: a² + b² = c²
Circle Circumference: C = 2πr

About This Calculator

This Related Rates Calculator helps you solve rate-of-change problems in calculus by applying the chain rule to geometric formulas like area, volume, and distance. It supports multiple problem types including area of circles, squares, rectangles, and triangles, as well as volume and motion-related formulas.

How It Works

Select a problem type and enter the known values, such as variable values and their time derivatives. The calculator differentiates the chosen formula with respect to time using the chain rule and outputs the target rate with detailed steps.

Circle Area:
A = \pi r^2 \rightarrow \frac{dA}{dt} = 2\pi r \frac{dr}{dt}
Square Area:
A = s^2 \rightarrow \frac{dA}{dt} = 2s \frac{ds}{dt}
Rectangle Area:
A = lw \rightarrow \frac{dA}{dt} = l \frac{dw}{dt} + w \frac{dl}{dt}
Triangle Area:
A = \frac{1}{2}bh \rightarrow \frac{dA}{dt} = \frac{1}{2}b \frac{dh}{dt} + \frac{1}{2}h \frac{db}{dt}
Sphere Volume:
V = \frac{4}{3}\pi r^3 \rightarrow \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}
Cylinder Volume:
V = \pi r^2 h \rightarrow \frac{dV}{dt} = \pi r^2 \frac{dh}{dt} + 2\pi rh \frac{dr}{dt}
Cone Volume:
V = \frac{1}{3}\pi r^2 h \rightarrow \frac{dV}{dt} = \frac{1}{3}\pi \left(2rh \frac{dr}{dt} + r^2 \frac{dh}{dt}\right)
Cube Volume:
V = s^3 \rightarrow \frac{dV}{dt} = 3s^2 \frac{ds}{dt}
Pythagorean Theorem:
a^2 + b^2 = c^2 \rightarrow \frac{dc}{dt} = \frac{a \frac{da}{dt} + b \frac{db}{dt}}{c}
Circle Circumference:
C = 2\pi r \rightarrow \frac{dC}{dt} = 2\pi \frac{dr}{dt}

Frequently Asked Questions

What is a related rates problem?

A related rates problem involves finding the rate at which one quantity changes by relating it to other changing quantities.

Do I need to use derivatives?

Yes, these problems require implicit differentiation of the formula with respect to time.

Which formulas are supported?

The calculator supports common formulas such as area of circles, rectangles, triangles, volume of cones, spheres, and motion/distance relationships.

Can I enter negative rates?

Yes, you can input negative values for variables or rates to indicate decreasing quantities.

What unit should I use?

You can use any consistent units, like meters and seconds or inches and minutes. The calculator reports result in squared or cubic units per time.

How accurate are the results?

The results are computed to high precision (up to 10 decimal places) and rounded to 6 by default.