Nine circles of radius 0.5 are externally tangent to a circle of radius 1 and also are tangent to one another.
Find the distance between the centers of the first and the last of the smaller circles.
The centers of two of the small tangent circles are 1 unit apart from one to the next. Half the angle subtended at the center of the large circle by the two adjacent centers is arcsin(1/3), so the full subtended angle is 2*arcsin(1/3). Going from the first to the last subtends eight times this, or 16*arcsin(1/3). The angular gap back to the first center is thus 360° - 16*arcsin(1/3) or 2*pi - 16*arcsin(1/3) when using radian measure.
The asked for distance between centers is twice 1.5 times the sine of half this gap, or 3*sin(180° - 8*arcsin(1/3)) in degree measure or 3*sin(pi - 8*arcsin(1/3)) in radian measure.
Numerically it's approximately 1.23121290478204 units.
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Posted by Charlie
on 2018-09-01 14:37:37 |
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