Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of interior points located in the polygon and the number b of boundary points placed on the polygon's perimeter: Image File history File links An illustration of Picks theorem, with 39 interior points and 14 boundary points. ... Look up polygon in Wiktionary, the free dictionary. ... The integers are commonly denoted by the above symbol. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... This article is about the physical quantity. ...

A = i + b/2 − 1.

In the example shown, we have i = 39 and b = 14, so the area is A = 39 + 14/2 − 1 = 39 + 7 − 1 = 45 (square units).

Note that the theorem as stated above is only valid for simple polygons, i.e., ones that consist of a single piece and do not contain "holes". For more general polygons, the "−1" of the formula has to be replaced with "−χ(P)", where χ(P) is the Euler characteristic of P. It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...

The result was first described by Georg Alexander Pick in 1899. The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra. Georg Alexander Pick (1859, Vienna – 1943) was an Austrian mathematician, after whom Picks theorem is named. ... Year 1899 (MDCCCXCIX) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Friday [1] of the 12-day-slower Julian calendar). ... In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them. ... For the game magazine, see Polyhedron (magazine). ...

Proof

Consider a polygon P and a triangle T, with one edge in common with P. Assume Pick's theorem is true for P; we want to show that it is also true to the polygon PT obtained by adding T to P. Since P and T share an edge, all the boundary points along the edge in common are merged to interior points, except for the two endpoints of the edge, which are merged to boundary points. So, calling the number of boundary points in common c, we have

i_PT = (i_P + i_T) + (c − 2)

and

b_PT = (b_P + b_T) − 2(c − 2) − 2.

From the above follows

(i_P + i_T) = i_PT - (c − 2)

and

(b_P + b_T) = b_PT + 2(c − 2) + 2.

Since we are assuming the theorem for P and for T separately,

Therefore, if the theorem is true for polygons constructed from n triangles, the theorem is also true for polygons constructed from n + 1 triangles. For general polytopes, it is well known that they can always be triangulated. That this is true in dimension 2 is an easy fact. To finish the proof by mathematical induction, it remains to show that the theorem is true for triangles. The verification for this case can be done in these short steps: In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In advanced geometry, in the most general meaning, triangulation is a subdivision of a geometric object into simplices. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

• directly check that the formula is correct for any rectangle with sides parallel to the axes;
• verify from that case that it works for right-angled triangles obtained by cutting such rectangles along a diagonal;
• now any triangle can be turned into a rectangle by attaching (at most three) such right triangles; since the formula is correct for the right triangles and for the rectangle, it also follows for the original triangle.

The last step uses the fact that if the theorem is true for the polygon PT and for the triangle T, then it's also true for P; this can be seen by a calculation very much similar to the one shown above. In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ...

External links

• Pick's Theorem (Java) at cut-the-knot
• Pick's Theorem