Distance travelled is speed integrated over time. That one sentence is the whole arc length formula.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The integral
Arc length from a to b is the integral of the magnitude of r′(t) dt. Everything hard is inside the square root.
The easy case
If speed is constant, the integral is just speed times elapsed time. No integration is needed at all.
Curvature
Curvature measures how fast the direction of travel turns per unit of distance. A straight line has curvature zero.
A circle
A circle of radius R has constant curvature 1/R. A small circle bends sharply; a huge one is nearly straight.
Radius of curvature
The reciprocal of curvature is the radius of the circle that best hugs the curve at that point.
Arc length integrates speed
The length of a curve is the integral of |r′(t)| over the parameter interval. Distance travelled is speed accumulated over time, which is what the formula literally says.
Length does not depend on parametrisation
Traversing the same curve faster changes the speed and the interval in compensating ways. The computed length is a property of the curve, not of how it is traced.
Curvature measures bending
It is the rate at which the unit tangent changes with respect to arc length. A straight line has curvature zero and a small circle has large curvature — tighter turns mean more curvature.
Why curvature is computed
Road and railway design bound curvature to limit lateral force, and computer graphics uses it to decide how finely to subdivide a curve. It is a practical quantity, not only a theoretical one.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
- Slope of the tangent0
Step 3: Watch an Example
One step at a time.
Watch Yusuf Measure a Helix
Yusuf finds the length of r(t) = ⟨3cos t, 3sin t, 4t⟩ from t = 0 to t = 2.
- Step 1
He differentiates to get r′(t) = ⟨−3sin t, 3cos t, 4⟩.
Step 4: Your Turn
Practice makes it stick.
The Track
Problem 1 of 2
A cart moves at constant speed 6 for 7 seconds. How far does it travel?
The Bend
Problem 2 of 2
A circular bend of radius 20 metres. What is its curvature, as a decimal?
Length and Bend
1 of 8
Speed is constant at 9 from t = 0 to t = 4. What is the arc length?
2 of 8
A circle of radius 4. What is its curvature, as a decimal?
3 of 8
A straight line. What is its curvature?
4 of 8
Curvature is 0.2. What is the radius of curvature?
5 of 8
r(t) = ⟨5t, 12t, 0⟩ from t = 0 to t = 3. What is the arc length?
6 of 8
Which bends more sharply, a circle of radius 2 or one of radius 10? Enter the radius of the sharper one.
7 of 8
Order these paths from least curved to most curved at their tightest point.
- 1A gentle motorway curve of radius 500 m
- 2A roundabout of radius 20 m
- 3A hairpin bend of radius 5 m
- 4A straight road
8 of 8
The speed of a particle is 0 for a whole interval. What is the arc length over that interval?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A circle of radius 25. What is its curvature, as a decimal?
Question 2 of 2
What is being integrated in an arc length integral?
What You Learned
- Arc length is the integral of speed, because distance is speed accumulated over time.
- Constant speed turns the integral into a multiplication.
- Curvature is how fast direction turns; a circle of radius R has curvature 1/R.