A traditional soccer ball has 20 regular (spherical) hexagons with 12 regular pentagons situated so each is surrounded by five of the hexagons.
Given such a ball that's 22 cm in diameter, what is the arc length of one side where two polygons meet (penta/hexa or hexa/hexa) in centimeters?
The circumference of the ball is 22pi cm. One way of drawing a circumference goes in the following order:
HHPEPHHPEP
a=arc length
H=hexagon height through two parallel sides: a*sqrt(3)
P=pentagon height: a*0.5sqrt(5+2sqrt(5))
E=edge separating two hexagons: a
These are based on the plane figure lengths, but the proportions stay the same when puffed into a circle.
So we have a*(2 + 4sqrt(3) + 2sqrt(5+2sqrt(5)) = 22pi cm
a=4.582 cm
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Posted by Jer
on 2024-06-28 12:59:07 |
Solution
