Related Rates Triangle Calculator
Calculate the rate of change of the hypotenuse ($dc/dt$) based on the lengths and rates of the legs.
Hypotenuse Length ($c$): units
Rate of Hypotenuse ($dc/dt$): units/sec
Understanding Related Rates in Right Triangles
In calculus, "Related Rates" problems involve finding the rate at which one quantity changes by relating that quantity to others whose rates of change are known. For a right-angled triangle, we use the Pythagorean Theorem as our primary equation.
Equation: a² + b² = c²
Derivative with respect to time (t):
2a(da/dt) + 2b(db/dt) = 2c(dc/dt)
Derivative with respect to time (t):
2a(da/dt) + 2b(db/dt) = 2c(dc/dt)
By simplifying the derivative (dividing by 2), we get the core formula used in this calculator:
dc/dt = (a(da/dt) + b(db/dt)) / c
Real-World Example: The Sliding Ladder
Imagine a 5-meter ladder leaning against a wall. If the bottom of the ladder (Side $a$) is 3 meters from the wall and is being pulled away at a rate of 2 m/s, how fast is the top of the ladder (Side $b$) sliding down?
- Side $a$: 3 meters
- Side $c$ (Ladder length): 5 meters (constant, so $dc/dt = 0$)
- $da/dt$: 2 m/s
Using the calculator above, you can solve for any unknown rate. Note that if a side is shrinking (like a ladder sliding down), the rate should be entered as a negative number.
Common Variables Explained
- Side $a$ and $b$: The horizontal and vertical legs of the triangle.
- Side $c$: The hypotenuse (the diagonal side).
- $da/dt$: How fast side $a$ is growing or shrinking over time.
- $db/dt$: How fast side $b$ is growing or shrinking over time.
- $dc/dt$: The resulting speed at which the hypotenuse length is changing.
How to Use This Calculator
- Enter the current length of side $a$ and its rate of change ($da/dt$).
- Enter the current length of side $b$ and its rate of change ($db/dt$).
- If a side is staying constant (like a fixed ladder length), its rate of change is 0.
- Click "Calculate" to find the current length of the hypotenuse and the speed at which it is changing.