Inscribed and Circumscribed Figures
Inscribed and Circumscribed Figures is a way to estimate a circle's size by drawing shapes inside and around it.
A round cookie cutter sits on a square piece of dough. The square is "circumscribed" — it wraps around the circle on the outside. Now press a smaller square into the dough inside the circle. That inner square is "inscribed" — it fits inside the circle. Both squares help you estimate how much dough the circle covers.
Explaining inscribed and circumscribed figures by grade level
Draw a square inside a circle so all four corners touch the edge. Now draw a bigger square around the circle so all four sides touch it. The circle fits between the two squares. The more sides your shapes have, the closer they hug the circle.
Projects that explore inscribed and circumscribed figures
Drawing shapes around a circle gives an upper estimate of its size. You start with a square drawn around a circle of radius 1, then double the number of sides to 8, then 16, and so on. As each polygon gains more sides, its perimeter drops closer to the true circumference. This process produces a recursive equation that approaches pi from above — and it turns out to match a famous upper-bound expression published by Francois Viete centuries ago.
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.