Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Curvature measures how sharply a curve bends; a small circle bends hard, a big one barely bends.
Common stuck point:The procedure for curvature intuition is the easy part; the trap is saying larger radius gives more curvature. Asking "Am I measuring how sharply a curve bends, not just its length or position?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Am I measuring how sharply a curve bends, not just its length or position?
Worked Examples
Example 1
easy
A circle has radius r=4 cm. What is its curvature κ? Compare with a circle of radius r=1 cm.Circle of radius r = 4 has curvature κ = 1/r = 1/4
Answer
κ=0.25 cm−1 for r=4; κ=1 cm−1 for r=1. Smaller circles curve more.
First step
1
Step 1: For a circle, curvature κ=r1.
Full solution
2
Step 2: For r=4: κ=41=0.25 cm−1.
3
Step 3: For r=1: κ=11=1 cm−1.
4
Step 4: The smaller circle (r=1) has curvature 4× greater, meaning it bends more sharply.
Curvature κ=1/r measures how sharply a curve bends. A large circle is nearly flat (low curvature), while a small circle bends tightly (high curvature). A straight line has radius of curvature ∞ and curvature 0.
Example 2
medium
Two circular arcs lie along a road: arc A has radius 200 m (gentle bend) and arc B has radius 50 m (sharp bend). Calculate the curvature of each and explain which is safer to drive at high speed.
Example 3
easy
Explain why a roundabout with radius 15 m is curvier than a freeway curve with radius 300 m.
Example 4
medium
Why does an ellipse NOT have constant curvature, while a circle does?
Example 5
medium
A road designer wants a constant centripetal-feel curve. Which shape should the road follow and why?
Example 6
hard
Why does a railroad use a 'clothoid' (spiral) transition between straight track and a circular curve?
Example 7
challenge
A sphere of radius R=2 has uniform curvature on its surface. Why can't you flatten this sphere onto a plane without distortion?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
A coin has diameter 2.4 cm. What is the curvature of its edge?Coin edge: diameter 2.4 cm, so r = 1.2 cm — find curvature κ = 1/r
Example 2
hard
The osculating circle (circle of curvature) at a point on a curve has radius ρ. If the curvature at point P is κ=0.4 cm−1, find ρ. If the curvature doubles, what happens to ρ?Osculating circle: κ = 0.4 cm⁻¹ — find the radius of curvature ρ = 1/κ
Example 3
easy
What is the curvature of a straight line?
Example 4
easy
A circle has radius 4. What is its curvature?Curvature κ = 1/r = 1/4 for a circle of radius 4
Example 5
easy
Which curves more sharply: a circle of radius 2 or a circle of radius 10?
Example 6
easy
Does a tight hairpin turn have high or low curvature?
Example 7
easy
As a circle's radius grows very large, what happens to its curvature?
Example 8
easy
A circle has curvature 61. What is its radius?Given κ = 1/6, find radius r = 1/κ
Example 9
easy
Which has greater curvature: the edge of a dinner plate or the edge of a coin?
Example 10
easy
Does curvature stay the same all around a circle?
Example 11
medium
Why does a larger circle have SMALLER curvature, even though it's bigger?
Example 12
medium
A road curve must have curvature at most 501 per meter for safety. What is the minimum allowed turning radius?Road safety limit κ = 1/50 m⁻¹ — what is the minimum turning radius r?
Example 13
medium
Does an ellipse have constant curvature?
Example 14
medium
Two circular arcs have radii 3 and 12. What is the ratio of their curvatures (radius 3 to radius 12)?
Example 15
medium
At a point on a curve, the 'best-fitting circle' has radius 5. What is the curve's curvature there?Osculating (best-fitting) circle has radius r = 5 — find κ = 1/r
Example 16
medium
A car turns a corner of radius 25 m, then a sharper corner of radius 10 m. Which corner has greater curvature?
Example 17
medium
Why does the curvature of a curve relate to the tangent line's rate of turning?
Example 18
medium
Order from least to greatest curvature: a straight road, a gentle highway curve (r = 500 m), a roundabout (r = 20 m).
Example 19
challenge
Why does a highway use a gradual 'spiral' transition between a straight section and a circular curve, in terms of curvature?
Example 20
challenge
The curvature of a curve at a point is 41. A car of length negligible travels it at speed v. The sideways (centripetal) acceleration is κv2. Find the radius of curvature and express the acceleration in terms of v.
Example 21
challenge
Explain why a straight line can be seen as a circle of 'infinite radius', using curvature.
Example 22
challenge
A sphere of radius R — how does its surface curvature compare to a circle of radius R, qualitatively, and why does this matter for flat maps of Earth?
Example 23
easy
A circle has radius 5 m. What is its curvature?
Example 24
easy
Which has greater curvature: a circle of radius 7 or a circle of radius 0.5?
Example 25
easy
A bicycle wheel has radius 0.3 m. What is the curvature of the rim?
Example 26
easy
True or false: doubling a circle's radius halves its curvature.
Example 27
medium
A satellite orbits Earth at altitude where the orbit radius is 7000 km. What is the curvature of the orbit?
Example 28
medium
Order from least to greatest curvature: a coin (radius 1 cm), a basketball hoop ring (radius 23 cm), a soccer field's center circle (radius 9.15 m).
Example 29
medium
A skateboard half-pipe has a quarter-circle cross-section of radius 2.5 m. Find the curvature of the surface profile.
Example 30
medium
A turning radius restriction posts ρ≥50 m. What is the largest allowed curvature?
Example 31
medium
At a point on a curve, the best-fitting (osculating) circle has radius 2 m. What is the curvature there?
Example 32
medium
A circle has circumference 20π m. Find its curvature.
Example 33
hard
At a sharp bend, the osculating circle has radius 0.25 m. The car's speed is 5 m/s. Find the centripetal acceleration using a=κv2.
Example 34
hard
A curve has curvature r1 at a point and the tangent line makes angle θ with the x-axis. If you walk along the curve a tiny arc length s, by how much does the tangent angle change?
Example 35
hard
A planet has a roughly spherical shape with radius 6400 km. What is the curvature of a great-circle path on its surface (in inverse km)?
Example 36
hard
A truck's safe maximum centripetal acceleration is 3 m/s2. At speed 15 m/s, what is the maximum curvature the road can have?
Example 37
challenge
Two arcs join smoothly at a point. Arc 1 has radius 40 m, arc 2 has radius 60 m. By how much does the curvature drop across the join?