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Corners of a square or triangle cut off to form a regular octagon or hexagon

This type of question, asked 3 times: CDS 2 2019, CDS 2 2023, CDS 2 2026. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2019

    76
    The corners of a square of side 'a' are cut away so as to form a regular octagon. What is the side of the octagon ?

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  2. Question 2

    CDS 2 2023

    54
    The corners of an equilateral triangular plate were cut in such a manner that it forms a regular hexagonal plate. What is the ratio of the area of the triangular plate to the area of the hexagonal plate ?

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  3. Question 3

    CDS 2 2026

    59
    A square of side length 4 cm has its corners cut away in such a manner so as to form a regular octagon. What is the length of the side of the octagon ? (Take 2\sqrt{2} = 1.41)

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