Curvature Intuition Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy

A circle has radius

r = 4

cm. What is its curvature

\kappa

? Compare with a circle of radius

r = 1

cm.

Solution

1
Step 1: For a circle, curvature $\kappa = \dfrac{1}{r}$ .
2
Step 2: For $r = 4$ : $\kappa = \dfrac{1}{4} = 0.25$ cm $^{-1}$ .
3
Step 3: For $r = 1$ : $\kappa = \dfrac{1}{1} = 1$ cm $^{-1}$ .
4
Step 4: The smaller circle ( $r=1$ ) has curvature $4\times$ greater, meaning it bends more sharply.

Answer

\kappa = 0.25

^{-1}

for

r=4

;

\kappa = 1

^{-1}

for

r=1

. Smaller circles curve more.

Curvature

\kappa = 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