Because pi equals a circle's edge length divided by its width, any polygon drawn inside or around a circle gives an approximate value. Ancient math used exactly this approach — inscribed and circumscribed shapes whose perimeters bracket the true value. The more sides the polygon has, the closer the estimate gets. You can track how fast the improvement happens by comparing errors from polygons with different side counts across six different pairs. The error ratio follows a clear pattern: as the side count grows, it approaches the square of the inverse of the side-count ratio, meaning the error shrinks in a predictable, measurable way.

Shapes drawn inside and around a circle act as lower and upper bounds for pi — one flat outline sits just within the curve, the other just outside it. As you test six different pairs and track how the error ratio changes, a clear pattern emerges: the gap between the two estimates shrinks in a predictable way. It approaches the square of the inverse of the side-count ratio, meaning each doubling of sides yields a reliably smaller error.

Every circle's circumference connects to pi through the fixed ratio C = πd, and ancient mathematicians narrowed in on that value by drawing polygons inscribed in and around circles. The more sides a polygon has, the closer its perimeter gets to the true circumference. You calculate pi by averaging the perimeters of polygons drawn inside and around a circle, and the error ratio follows a clear pattern — shrinking in a predictable way as you add more sides.

Ancient mathematicians estimated pi by drawing polygons inside and around circles — and the more sides a polygon has, the closer the estimate gets. You calculate pi by averaging the perimeters of polygons inscribed in and circumscribed about a circle, then track how the error changes as the side count grows. Testing six different pairs of polygons, a clear pattern emerges: the error ratio approaches the square of the inverse of the side-count ratio. That means the error shrinks in a predictable way as you add more sides, giving you a precise measure of how fast the shape converges toward a perfect circle.

Ancient math used shapes drawn inside and around circles to estimate pi. When you inscribe a polygon within a circle and circumscribe another around it, you get estimates from both sides. You calculate pi by averaging their perimeters, then track how the error changes as the side count grows. Testing six different pairs reveals a clear pattern: the error ratio approaches the square of the inverse of the side-count ratio. That means as you add more sides, the error shrinks in a predictable, measurable way.

Ancient mathematicians estimated pi by drawing polygons inside and around circles, then measuring the total distance around each shape. You do the same: average the perimeters of polygons inscribed in and circumscribed about a circle to get your pi estimate. As you increase the number of sides, each perimeter traces the circle's edge more closely, and the estimate improves. When you test six different polygon pairs and track how the error ratio changes, a clear pattern emerges — the error shrinks in a predictable way as the side count grows.

How far does your estimate of pi land from the true value? In this experiment, you use polygon perimeters to estimate pi, then calculate the gap between your result and actual pi. As the polygon gains more sides, that gap shrinks — and it shrinks in a predictable pattern. The error ratio approaches the square of the inverse of the side-count ratio, meaning each increase in sides compresses the error by a calculable factor.