Pick's Theorem

What’s going on

Take squared paper and mark the corners of the squares: a grid of dots, one unit apart. Draw any polygon whose corners are on dots. Its sides can be slanted, and the shape can have dents. What is its area?

You could cut it into triangles and use formulas. Or you could just count:

• I, the number of dots strictly inside the shape;
• B, the number of dots on its boundary, corners included.

Then

area = I + B ÷ 2 − 1.

That is Pick’s theorem, published by the Austrian mathematician Georg Pick in 1899.

The pentagon in the scene has 23 dots inside (orange) and 12 on its edges (blue). Pick’s formula gives 23 + 6 − 1 = 28. Next to it the scene also works out the area with the shoelace formula, which uses only the coordinates of the corners and knows nothing about dots. It also says 28.

Drag a corner and watch both numbers change together. Press Clear and draw your own shape: click dots to place corners, then click the first corner again to close it.

Checking the pentagon by hand

Draw the smallest rectangle around the pentagon: it is 7 by 6, so 42 squares. What is left over is four right-angled triangles at the corners, with areas 1.5, 4, 4 and 4.5. Together that is 14, and 42 − 14 = 28.

Why might it be true?

Here is a way to see where each piece of the formula comes from. Imagine each dot owns a little square of land around it, one unit in area. A dot deep inside the shape owns all of its land inside the shape. A dot on a straight part of the edge has about half its land inside. So a first guess for the area is I + B ÷ 2.

The corners spoil this a little. Going all the way round the shape, the boundary turns through one full turn, and those turns take away exactly one unit of land in total. That is the − 1. (A full proof cuts the polygon into triangles and checks that the formula adds up correctly when shapes are glued together.)

When it breaks

Choose Bow tie (crossed) in the scene. Two sides cross, so this is not a proper polygon, and the formula gives nonsense: the dots say 17, while the shoelace formula, whose two halves cancel out, says 0. Pick’s theorem needs a simple polygon: its sides must not cross or touch, and it must have no holes. The scene warns you when sides cross.

How to count boundary dots fast

On a side that goes a steps across and b steps up, the dots are evenly spaced, and there are as many gaps between them as the greatest common divisor of a and b. A side going 6 across and 4 up has gcd(6, 4) = 2 gaps, so one dot in the middle. Add the gaps of all the sides and you get B.

Try it yourself

1. No dots inside

A lattice polygon has no dots inside and 10 dots on its boundary. What is its area? Can you draw one?

Answer

0 + 10 ÷ 2 − 1 = 4. For example, a 1 × 4 rectangle: it has 10 dots on its edges and none inside, and its area is 4.

2. The smallest triangle

A lattice triangle has no dots at all except its three corners. Show that its area is always ½, however long and thin it is.

Answer

I = 0 and B = 3, so the area is 0 + 1.5 − 1 = ½. Choose Thinnest triangle in the scene: it stretches across nine columns but still has area ½.

3. No equilateral triangles

Can you put the three corners of an equilateral triangle on dots of the square grid?

Answer

No. By Pick's formula the area of any lattice polygon is a whole number or a half. But an equilateral triangle with side s has area s² × √3 ÷ 4, and s² is a whole number (by Pythagoras, it is a sum of two squares). A whole number times √3 is never a whole number or a half, so such a triangle cannot exist.

Go deeper

• Georg Pick (1859–1942) worked in Prague. He died in the Theresienstadt concentration camp in 1942. His formula became widely known only decades after he published it.
• There is no such formula in three dimensions. In 1957 John Reeve found tetrahedra whose only lattice points are their four corners, yet whose volumes can be as large as you like.