Triangles in squares

Abstract

In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

102.51 Archimedes' quadrature of the parabola

“…QN NM arc MQ = 4 3 A 1 PMQ = 8 3 A 1 = 4 3 A PMQ = A Using Archimedean slices parallel to the -axis, we find that and the area of segment is . Elegant, it is not!…”