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This research explores the conditions under which isosceles right triangles can cover a rectangle through translative methods.

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April 15, 2026Open Access

Translative Covering a Square with Isosceles Right Triangles

Key Points

This research explores the conditions under which isosceles right triangles can cover a rectangle through translative methods.
Investigated the covering of a rectangle using homothetic copies of right isosceles triangles.
Analyzed area requirements for successful coverage of squares and rectangles.
Established strict bounds for the area of triangles needed for coverage.
Any collection of equal triangles with a total area of at least 2 allows for a 1×1 covering.
A total area of at least 3 from positive homothetic triangles also enables a 1×1 covering.
For rectangles with dimensions satisfying a≥5+334b, an area of at least 12(a+b)² ensures coverage.

Abstract

A translative covering the rectangle a×b with homothetic copies of a right isosceles triangle T (of the legs parallel to the sides of a×b) is considered. It is shown that any collection of equal triangles homothetic to T with the total area at least 2 permits a translative covering of 1×1; this bound is tight. It is also demonstrated that any collection of positive homothetic copies of T with the total area at least 3 permits a translative covering of 1×1. Moreover, it is proven that if a≥5+334b, then any collection of triangles homothetic to T with the total area at least 12(a+b)2 permits a translative covering of a×b; this bound is tight.

Translative Covering a Square with Isosceles Right Triangles

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Abstract

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