AMC 10 · 2006 · #6
Grade 7 geometry-2d
Pick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The perimeter is one curvy loop, but it is built from four identical semicircular arcs. Break the problem into one small piece: find the length of a single arc, then multiply by four. Finding one arc needs the circle circumference formula, which first needs the radius from the given side.
Side is the diameter
Each semicircle takes a whole side as its diameter, so the diameter is 2/π and the radius is half of it: 1/π.
The radius is always half the diameter, and here the diameter is just the square's side.
7.G.B.4Visualize Spatial RelationshipsLength of one arc
A semicircular arc is half of 2πr, namely πr, and r = 1/π makes the π cancel, so one arc has length 1.
Half a circle's edge is π times the radius, and the chosen radius makes the π cancel cleanly.
Half a circle's edge is pi times the radius, and the chosen radius makes the pi cancel cleanly.
▸ Why?
A whole circle's edge is two pi times its radius, which sets the size of any part of it.
▸ Why?
An arc is a fixed share of the whole edge, so half the turn gives half the length.
Add the four arcs
The boundary is exactly four such arcs, one per side, so the perimeter is 4·1 = 4, choice (D).
Four equal arcs make a perimeter four times the length of one arc.
3.OA.C.7Identify SubproblemsWhen a boundary is made of identical curved pieces, measure one piece and multiply, and a cleverly chosen radius can make the π cancel to a whole number.
- Side is the diameter
- Length of one arc
- Add the four arcs