![]() ![]() % of hexagon covered by circles = (3$\pi$/10.38) x 100 = 90.8% (rounded) Notice how not using exact values for the height of the triangles in the hexagon calculation means this is 0.1 cm 2 off - a type of rounding error introduced by rounding off numbers before the answer. radius of circle = 1cm so area = $\pi$ cm 2 and area of 3 circles = 3$\pi$ cm 2 There are 3 whole circles in each hexagon. We worked out the percentage of each plane covered by circles, by dividing the two patterns into hexagons and squares (see the diagrams above) We worked out the percentage of each hexagon covered by circles and the percentage of squares covered by circles.Ī hexagon is 6 triangles area of 1 triangle: base = 2cm height = 1.73cm so area of 1 triangle = 1.73cm 2 and the area of the hexagon is 10.38cm 2. Here is another method for Circle Packaging from Suzanne and Nisha of the Mount School, York: The hexagon touches the circle at the midpoints of its sides, the distance between the midpoints of opposite sides is $2$ units, the lengths of the sides of the hexagon are $1/\sqrt$. ![]() In the other case the packing of the plane can be produced by a tessellation of hexagons (like a honeycomb). Therefore the proportion of the plane covered by the circles is $\pi/4 = 0.785398\ldots = 78.5\%$ to 3 significant figures. The area of the circle is $\pi$ and the area of the square is $4$ square units. Let us say that the radius of the circle is $1$ unit. In either case the calculation of areas is the In one case it could be a tessellation of squares, either surrounding the circles or formed by joining the centres of the circles. To find the percentage of the plane covered by the circles in each of the packings we must find, within the original pattern, a shape that tessellates the plane and in each case this can be done in different ways. James of Christ Church Cathedral School, Oxford and Alexander from Shevah-Mofet School, Israel sent very good solutions to this question. ![]()
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