Count the dots, skip the ruler: a polygon's area hiding in plain sight
Draw any polygon on graph paper so that every corner lands exactly on a grid dot, no measuring allowed. Count the dots strictly inside the shape, then count the dots on its boundary, including the corners. Is there a simple formula connecting those two counts to the polygon's exact area?
Reveal the answer
Yes: area equals the number of interior dots, plus half the number of boundary dots, minus one. Georg Alexander Pick proved this in 1899, and it works for any simple polygon with grid-point corners no matter how irregular, letting you compute an exact area just by counting dots, with no measuring, no trigonometry, and no limit on how strange the shape's outline gets.