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MATH / Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere wh…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Six spheres of radius 11 are positioned so that their centers are at the vertices of a regular hexagon of side length 22. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
Plain-text mathematical notation (without MathML)
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon.  An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere.  What is the radius of this eighth sphere?
Original LaTeX notation
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon.  An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere.  What is the radius of this eighth sphere?

level

Level 5

type

Geometry

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Source and history

task_id
5f456c9e-dc5c-5222-8488-4e6abeab7699
task_key
geometry--test--5f456c9e-dc5c-5222-8488-4e6abeab7699
task_revision_id
1
config / split
geometry / test

Official source

initial import