Depth · one concept from first principles · step 2 of 4
The figure on the right shows circles, each with a radius of , drawn so that each circle passes through the center of the next one. How long is segment GN, in centimeters?
Along GN, G to the first center is one radius (5 cm); each center to the next is one more radius, and so is the last center to N.
The center-to-edge distance of any circle is its radius, so every piece of GN is the same 5 cm length.
3.G.A.1Draw A DiagramEvery piece of segment GN is the same length, one radius (5 cm): from G to the first center, each center to the next center, and the last center to N.
GN is exactly those pieces laid end to end with no gaps and no overlaps, so the length of GN is just the pieces added together.
Each piece is one radius long: the picture is drawn so every circle passes through the next center, which puts that next center right on the circle's rim, and G and N are the rim points at the two far ends.
A point sitting on a circle's rim is exactly one radius away from that circle's center, because that fixed center-to-rim distance is what the radius is.
Those radii are all the same 5 cm because the 21 circles are equal, so each piece equals a radius and each radius equals 5 cm, which makes every piece 5 cm.
Small cases: 2 circles → 3 pieces (2+1), 3 circles → 4 pieces (3+1), so 21 circles → 22 equal pieces of 5 cm.
There is always one more gap than the number of jumps between centers, because we add the two end radii.
3.OA.A.3Solve An Easier Related ProblemThere are 22 equal pieces, each 5 cm long: 22 × 5 = 110.
Repeated equal lengths join into a total by multiplication, a Grade 3 fact within 100.
3.OA.C.7Look For A PatternEvery gap is one radius, so just count the gaps and multiply -- only Grade 3 multiplication you already know!
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Path Length Comparison & Shortest Path. Why the shapes repeat →