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Equilateral triangle: circumradius, inradius or centroid distance from side or area

This type of question, asked 2 times: 8 Aug 2022 Tier II, 6 Mar 2023 Tier II. All 4 questions below, oldest first.

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  1. Question 1

    8 Aug 2022 Tier II

    15
    ABC is an equilateral triangle. If the area of the triangle is 36336\sqrt{3} , then what is the radius of circle circumscribing the triangle ABC?

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  2. Question 2

    8 Aug 2022 Tier II

    92
    ABC is an equilateral triangle with side 12 cm. What is the length of the radius of the circle inscribed in it?

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  3. Question 3

    8 Aug 2022 Tier II

    65
    PQR is an equilateral triangle and the centroid of triangle PQR is point A. If the side of the triangle is 12 cm, then what is the length of PA?

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  4. Question 4

    6 Mar 2023 Tier II

    17
    The centroid of an equilateral triangle PQR is L. If PQ = 6 cm, the length of PL is:

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