Related Rates- Triangle with Increasing Sides Problems
Related Rates: Triangle with Increasing Sides Problems
If you're taking calculus, related rates problems show up on nearly every exam. They test whether you understand derivatives—not just mechanically, but as rates of change that connect to each other. Triangle problems are among the most common. Here's how to solve them without losing your mind.
What Related Rates Actually Are
Related rates problems involve two or more quantities that change over time, connected by some relationship. You know how fast one quantity changes. You need to find how fast another changes.
The relationship between the quantities is usually geometric—area, volume, the Pythagorean theorem, or similar. The relationship between the rates of change comes from differentiating that geometric relationship with respect to time.
That's the whole concept. Everything else is execution.
The Standard Setup for Triangle Problems
Most triangle related rates problems fall into two categories:
- Right triangles — where the Pythagorean theorem gives you the relationship
- Any triangle — where area formulas, law of cosines, or trigonometry apply
Right triangles are simpler, so start there.
The Step-by-Step Method
Here's the process that works every time:
- Draw a diagram. Label the sides and angles. Name what changes with letters.
- Identify what you know and what you need. Write down the given rate (d-something/dt) and the unknown rate.
- Write the geometric relationship. For right triangles, this is usually a² + b² = c².
- Differentiate with respect to time. This is where most people mess up—remember the chain rule.
- Plug in the known values. Solve for the unknown rate.
Why the Diagram Matters
Don't skip it. A diagram keeps your variables straight. In triangle problems, sides are often labeled with letters, and the hypotenuse gets its own variable. If you mix these up, your equation will be wrong and your answer will be too.
Classic Example: The Ladder Problem
A 10-foot ladder leans against a wall. The bottom slides away at 3 ft/s. How fast does the top slide down when the bottom is 6 feet from the wall?
Step 1: Set Up
Let x = distance from wall to ladder's bottom. Let y = height on wall. The ladder length is constant at 10 feet.
Step 2: The Relationship
x² + y² = 10²
Step 3: Differentiate
2x(dx/dt) + 2y(dy/dt) = 0
Simplify: x(dx/dt) + y(dy/dt) = 0
Step 4: Find the Known Values
dx/dt = 3 ft/s (given). x = 6 ft.
Find y: 6² + y² = 100 → y² = 64 → y = 8 ft
Step 5: Solve
6(3) + 8(dy/dt) = 0
18 + 8(dy/dt) = 0
dy/dt = -18/8 = -9/4 ft/s
The negative sign means y is decreasing—the top of the ladder is sliding down. Don't forget that sign.
Another Common Type: Expanding Equilateral Triangle
Each side of an equilateral triangle increases at 2 cm/s. How fast does the area increase when each side is 10 cm?
Step 1: Set Up
Let s = side length. Area A = (√3/4)s²
Step 2: Differentiate
dA/dt = (√3/4)(2s)(ds/dt)
dA/dt = (√3/2)s(ds/dt)
Step 3: Plug In
s = 10, ds/dt = 2
dA/dt = (√3/2)(10)(2) = 10√3 cm²/s
This one is straightforward—no Pythagorean theorem, just the area formula.
Comparing Common Triangle Related Rates Scenarios
| Problem Type | Geometric Relationship | Key Formula |
|---|---|---|
| Right triangle (ladder) | Pythagorean theorem | a² + b² = c² |
| Equilateral triangle expanding | Area formula | A = (√3/4)s² |
| Isosceles triangle with changing angle | Trigonometry | A = (1/2)ab·sin(C) |
| Shadow problems | Similar triangles | Proportional sides |
Getting Started: Your Checklist
Before you start solving any triangle related rates problem:
- Draw the triangle and label everything
- Identify which quantities are changing and which are fixed
- Write the geometric formula that connects them
- Confirm which rate is given and which you need to find
- Differentiate carefully—every variable gets multiplied by its rate
- Substitute numbers only after differentiating
- Check your units—they should match
Where People Screw Up
Forgetting to Differentiate
Some students write the geometric formula and then just plug in numbers. That works for static geometry problems. Here, you must differentiate. The derivative is what connects the rates.
Chain Rule Errors
When differentiating something like x², you get 2x(dx/dt). When differentiating something like (area)², you get 2(area)(dA/dt). The chain rule applies every time a variable is a function of time.
Mixing Up Variables
Labeling is critical. If x is the base and y is the height, don't switch them mid-problem. Keep your diagram in front of you.
Solving for the Wrong Variable
Read the question twice. Sometimes you're asked for the rate of change of area, sometimes for a side length's rate of change. Make sure you're solving for what was actually asked.
Practical Approach for Any Triangle Problem
When you encounter a new problem you haven't seen before:
- Determine the shape—right triangle, equilateral, isosceles, or general
- Identify the relevant geometric relationship (area, Pythagorean theorem, trig)
- Express everything in terms of variables
- Take the derivative
- Plug in the given rate and the point where evaluation occurs
- Solve algebraically
This framework works for any triangle configuration. The specific formula changes; the process doesn't.
The Bottom Line
Related rates triangle problems are mechanical once you understand the setup. The geometry tells you the relationship. The calculus tells you how the rates connect. Differentiate the geometric formula, substitute your known values, and solve. That's it.
Practice the ladder problem until it's automatic. Then practice the expanding triangle. Those two cover most of what you'll see on exams.