Daily Math Puzzle: 2026-06-24
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2026-06-24
The ancient astronomers of Xylos observed a rare celestial event: three stars forming a perfect equilateral triangle. Inspired, their high priest decreed a new sacred temple be built on Earth, a vast, perfectly circular platform. Within this grand platform, three equally sized, perfectly circular sacred pools were to be constructed. The pools had to be placed such that each pool touched the other two, and all three also touched the outer edge of the main temple platform. If the radius of each small sacred pool is 'r', what is the radius of the large circular temple platform?
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Solution
r * ( (2 * sqrt(3) + 3) / 3 ) — Let R be the radius of the large circular temple platform and 'r' be the radius of each small sacred pool.
1. **Arrange the small pools:** Since the three small pools are equally sized and each touches the other two, their centers form an equilateral triangle. The distance between the centers of any two touching circles is the sum of their radii, so the side length of this equilateral triangle is r + r = 2r.
2. **Find the center of the large platform:** The large circular platform is tangent to all three small pools. Due to the symmetry of the arrangement, the center of the large circular platform will coincide with the geometric center of the equilateral triangle formed by the centers of the small pools.
3. **Calculate distance from center to a small pool's center:** For an equilateral triangle with side length 's' (in this case, s = 2r), the distance from its center to any of its vertices (which are the centers of the small pools) is given by the formula (s / sqrt(3)).
So, the distance from the center of the large platform to the center of any small pool is (2r / sqrt(3)). To rationalize the denominator, we multiply the numerator and denominator by sqrt(3): (2r * sqrt(3)) / 3.
4. **Determine the large platform's radius:** The radius R of the large platform is this calculated distance plus the radius of one of the small pools (since the large circle touches the small pool's outer edge).
R = (distance from center to small pool's center) + r
R = (2r * sqrt(3)) / 3 + r
Factor out 'r':
R = r * ( (2 * sqrt(3)) / 3 + 1 )
To combine the terms in the parenthesis, find a common denominator:
R = r * ( (2 * sqrt(3) + 3) / 3 )
Thus, the radius of the large circular temple platform is r * ( (2 * sqrt(3) + 3) / 3 ).
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