In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), (also Cauchy-Crofton formula) is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.

Suppose ${\displaystyle \gamma }$ is a rectifiable plane curve. Given an oriented line ℓ, let ${\displaystyle n_{\gamma }}$(ℓ) be the number of points at which ${\displaystyle \gamma }$ and ℓ intersect. We can parametrize the general line ℓ by the direction ${\displaystyle \varphi }$ in which it points and its signed distance ${\displaystyle p}$ from the origin. The Crofton formula expresses the arc length of the curve ${\displaystyle \gamma }$ in terms of an integral over the space of all oriented lines:

The differential form

is invariant under rigid motions of ${\displaystyle \mathbb {R} ^{2}}$, so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the kinematic measure.

The right-hand side in the Crofton formula is sometimes called the Favard length.

In general, the space of oriented lines in ${\displaystyle \mathbb {R} ^{n}}$ is the tangent bundle of ${\displaystyle S^{n-1}}$, and we can similarly define a kinematic measure ${\displaystyle d\varphi \wedge dp}$ on it, which is also invariant under rigid motions of ${\displaystyle \mathbb {R} ^{n}}$. Then for any rectifiable surface ${\displaystyle S}$ of codimension 1, we have $${\displaystyle \operatorname {area} (S)=C_{n}\iint n_{\gamma }(\varphi ,p)\;d\varphi \;dp.}$$where$${\displaystyle C_{n}={\frac {1}{2\cdot |{\text{unit ball in }}\mathbb {R} ^{n-1}|}}={\frac {\Gamma {({\frac {n+1}{2}})}}{2\pi ^{\frac {n-1}{2}}}}}$$

Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment, it must equal some function of the segment's length. Because, again, the formula is additive over concatenation of line segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the unit circle.

The proof for the generalized version proceeds exactly as above.