2025 AMC 10B Problem 12

Problem 12 of 25IntermediateGeometry

The figure below shows an equilateral triangle, a rhombus with a

60^\circ

angle, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let

T,

R,

and

H,

respectively, denote the ratio in each case of the total area of the disks to the area of the enclosing polygon.

Which of the following is true?

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Solution

Take the triangle with side

s.

Three disks of radius

r

give

s = 2r(1 + \sqrt3),

T = \dfrac{3\pi r^2}{(\frac{\sqrt3}{4})s^2}

= \dfrac{(2\sqrt3 - 3)\pi}{2} \approx 0.73.

For the rhombus with side

a,

the two disks sit on the long diagonal

a\sqrt3 = 6r,

r = \tfrac{a}{2\sqrt3}

and

R = \dfrac{2\pi r^2}{(\frac{a^2\sqrt3}{2})} = \dfrac{\pi\sqrt3}{9} \approx 0.60.

For the hexagon with side

a,

each of the six disks touches a side at its midpoint, again giving

r = \tfrac{a}{2\sqrt3}

and

H = \dfrac{6\pi r^2}{(\frac{3\sqrt3}{2})a^2}

= \dfrac{\pi\sqrt3}{9} \approx 0.60.

H = R \lt T.

Therefore, the answer is C.