Reasoning · Grade 6-2 Circumference of a Circle
Problem
The square gives a diameter of 2 cm.
A circle squeezed into a square so tightly that it kisses every side has nowhere left to grow: its width is the square's width. Once you see the diameter in the picture, the radius is just half of it.
7.G.B.4Draw A DiagramThe hexagon's side equals the radius.
This is the one fact that makes the hexagon the friendliest polygon to inscribe: the angle-sum argument turns each of the six slices into an equilateral triangle, so the side you need is a length you already know - the radius - and no measuring or square roots are involved.
8.G.A.5Identify SubproblemsThe hexagon's perimeter is 6 cm.
Perimeter is just the walk around the outside; for a regular shape it is one side repeated, so a single multiplication finishes it.
3.MD.D.8Identify SubproblemsThe square's perimeter is 8 cm.
The square is the outside wrapper here, and its side is the diameter itself, so its perimeter is simply four diameters - no new measurement is needed.
4.MD.A.3Identify SubproblemsThe circle's rim lies between the two.
Nobody can measure a curve exactly with a ruler, but anyone can measure straight sides. Trapping the unknown curve between one shape you can measure that is definitely smaller and another that is definitely bigger turns an impossible measurement into two easy ones.
7.G.B.4Solve An Easier Related ProblemThe circumference is trapped between the hexagon's perimeter and the square's, because one sits inside the circle and the other outside.
An inside shape has a shorter boundary than the circle and an outside shape a longer one, so the two bounds chain around it.
Each polygon's perimeter is its straight sides laid end to end, so both bounds can be worked out without any curved measuring.
Dividing by the diameter puts pi between 3 and 4.
Two useful ideas meet here. First, a comparison survives division by the same positive amount, the way three children keep their order in a race if every distance is halved. Second, centimetres divided by centimetres leaves no unit at all, which is why pi is a plain number and comes out the same for a coin or for a running track.
7.EE.B.4Analyze The UnitsThe square's actual size was never needed.
Seeing the 3 as 'six radii fit around, and the diameter is two radii' explains why the lower bound is exactly 3 and not some messy number - and why more polygon sides always improve the estimate.
7.G.B.4Solve An Easier Related ProblemTrap the circle between a hexagon inside and a square outside, and you can prove pi is between 3 and 4 without ever being told what pi is.