Depth · one concept from first principles · step 2 of 4
The figure at the right is the top view of four identical cylinders packed snugly into a square box; each cylinder's base has a circumference of . What is the area of the empty space, in ? (Ignore the thickness of the box.)
The circumference is the radius times two times 3.14, so divide it back out.
Each circle has a radius of 7 cm.
7.G.B.4Work BackwardsTwo circles sit side by side along each edge, so the side is two diameters.
The square is 28 cm on a side.
6.G.A.1Draw A DiagramThe square's side is two diameters.
Two circles sit side by side along each edge, touching each other and touching the square, so the side runs from one circle's far edge to the other's without any gap.
Each circle contributes its own diameter to that run, because a circle's width across the centre is the same in every direction.
The square is its side multiplied by itself; each circle is its radius twice times 3.14, and there are four of them.
784 against 615.44 cm².
7.G.B.4Draw A DiagramWhat is empty is the square with the four circles taken out.
The empty space is 168.56 cm².
7.G.B.4Change Focus Count The ComplementWhen a leftover shape is awkward, measure everything else and subtract.
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