I know there are a lot of ways to calculate $\pi$. But is there a way to explain and calculate it together with a kid, using only basic math and geometry? So without trigonometric functions, square root, etc.?
Edit: sorry, maybe the question is confusing. I am aware of how to approximate the value by measuring inner and outer polygons. The question is really about getting close to $\pi$ with calculations and not measuring.
On
Get a piece of string and a tape measure. Measure the circumference and diameter of as many circular objects as you can find (tins, plant pots, ...). The ratio of the two is then $\pi$. Draw a graph of circumference against diameter if they are a bit older. The points will lie on an approximate straight line and the slope is a good measure of $\pi$.
Draw as close to a perfect circle as you can with a compass. Measure it's perimeter or area and divide by the diameter or the radius squared, respectively, to compute $\pi.$
To measure perimeter, try wrapping a string around the circle and cutting it. To measure the area, try cutting out the circle and weighing the piece of paper you printed on it. Or you could be like Archimedes if your kid knows some geometry: Compare the circle with an inscribed and circumscribed polygon, which hopefully you can calculate the area of more easily, to get an upper and lower bound.