21:I[344796,["/_next/static/chunks/0whdtqcc5h8md.js","/_next/static/chunks/1qo17jq3o72yb.js","/_next/static/chunks/00mfjk9z2ewie.js","/_next/static/chunks/3_0kqrj9zuodm.js","/_next/static/chunks/2q9x6spyz4qxv.js","/_next/static/chunks/2cxhk-z7t3ztw.js","/_next/static/chunks/35kg0tmkl08ow.js","/_next/static/chunks/09hukzrse-ioj.js","/_next/static/chunks/3-qfh2-yexhl7.js","/_next/static/chunks/0_-iwh06bu1or.js"],"default"] 22:Taf6,## Overview A regular hexagon `PQRSTU` sits inside a circle with all six vertices on the circle, and the shaded area is everything inside the circle but outside the hexagon. That region is the circle's area minus the hexagon's area, so the whole task is to turn the one given length — the shortest diagonal, `9` units — into the circle's radius and the hexagon's side. The answer is `27π - 81√3/2 square units`, approximately `14.67`. ## Key Insight A regular hexagon splits into six equilateral triangles meeting at its centre, so its circumradius equals its side: `R = s`. Its diagonals come in two lengths — the short ones join vertices two apart (`PR`, `QS`, ...) and measure `s√3`, while the long ones join opposite vertices (`PS`, `QT`, `RU`) and measure `2s`. Since `√3 < 2`, the shortest diagonal is `s√3`, and that is the `9` units the problem gives. ## Approach From `s√3 = 9` we get `s = 9/√3 = 3√3`, hence `s² = 27` and `R = s = 3√3`. The circle's area is `πR² = 27π`. The hexagon's area is six equilateral triangles of side `s`, that is `6 × (√3/4)s² = (3√3/2)s² = (3√3/2)(27) = 81√3/2`. Subtracting gives the shaded total `27π - 81√3/2`. ## Why This Works The six segments together are exactly the set difference between the disc and the hexagon, so subtraction counts each segment once and nothing else. The script `verify.py` in the work notes confirms the numbers two independent ways: symbolically it prints `side s = 3*sqrt(3)`, `circle area = 27*pi = 84.8230016469`, `hexagon area = 81*sqrt(3)/2 = 70.1480577065` and `shaded = 27*pi - 81*sqrt(3)/2 = 14.6749439404`; numerically it places six vertices at radius `3√3`, checks the two-apart distance is `9.0`, and gets the hexagon area by the shoelace formula as `70.1480577065`, so the same shaded value `14.6749439404` falls out. Comparing the options: `162π - 243√3 = 88.0497` is six times too large, `81π - 81√3/2 = 184.3209` uses the diagonal `9` as the radius while keeping the correct hexagon area, and `3π - 9√3/2 = 1.6305` comes from the slip `s² = 9/3 = 3` instead of `s² = 81/3 = 27`. Only `27π - 81√3/2` matches, so that is the answer. ## Complexity Constant work — a handful of arithmetic operations on the single given length, `O(1)` time and `O(1)` space. ## Common Pitfalls Reading `9` as the side length, or as the radius, inflates every area by a factor of three; the given `9` is the short diagonal `s√3`. Squaring carelessly is the other trap: `s = 9/√3` means `s² = 81/3 = 27`, not `9/3 = 3`. Some solvers compute a single segment and forget to multiply by six, and others use `2s` — the long diagonal — as the shortest one. Finally, keep the answer exact: rounding `π` and `√3` early loses the closed form the options are written in. 19:[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"learningResourceType\":\"Coding Question\",\"name\":\"Shaded Segments Around an Inscribed Hexagon\",\"description\":\"A regular hexagon PQRSTU is inscribed within a circle. 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A regular hexagon PQRSTU is inscribed within a circle, as shown in the attached figure.

In the figure, the six shaded regions are the parts of the circle that lie outside the hexagon — the circular segments cut off by the sides PQ, QR, RS, ST, TU and UP.

The length of the shortest diagonal of the regular hexagon is equal to 9 units.

What is the total area of the shaded regions?

(inferred — the option list shown in the original screenshot contained no expression equal to the exact area, so the weakest distractor 81π - 27√3 square units was replaced by the exact value; the other three options are kept verbatim from the source.)

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Shaded Segments Around an Inscribed Hexagon

Problem Statement

Question 1