Reasoning · Grade 3-2 Circles and Length
Problem
With small diameter d, the radii are d/2 and d.
Halving a diameter to get a radius is the same 'split into two equal parts' idea kids use with fractions of a whole.
3.NF.A.1Draw A DiagramThe side joining the two large circles is 2d.
Each tangent side is just two touching radii laid end to end, so adding them gives the side length.
3.OA.A.1Identify SubproblemsEach side of the triangle is two touching radii laid end to end, so its length is those two radii added.
Every point of a circle sits the same distance from its centre, so the piece from a centre to the touching point is that circle's radius wherever you look.
The two radii meet at the touching point with no gap and no overlap, so together they are exactly the whole side.
The other two are d + d/2 each, so the perimeter is 5d.
Perimeter is just walking all the way around, so I add up every side; the two half-circles' radii team up into one full d.
3.MD.D.8Identify Subproblems5d = 30 makes the small diameter 6 in.
Five equal groups make 30, so each group is 6 — basic division a third-grader can do.
3.OA.A.3Guess And CheckEvery side is just two touching radii added together, so the whole trip around the triangle is 5 small diameters — split 30 into 5 equal parts to get 6 in!