# Length of a side of an inscribed square in a triangle

## Description

Every acute triangle has three inscribed squares (squares in its interior such that all four of a square’s vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle’s right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle’s longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has side of length q and the triangle has a side of length a, part of which side coincides with a side of the square, then q, a, and the triangle’s area T are related

Related formulas## Variables

q | The length of the inscribed square (m) |

T | The triangle's area (m^{2}) |

a | The triangle's side (a part of which coincides with the side of the square) (m) |