rectangleja

Rectangleja is a geometric figure defined as a convex pentagon obtained by adjoining to one side of a rectangle a triangle that shares the entire side as its base. The rectangle has width w and height h; the attached triangle has base equal to w and height t, measured perpendicular to the base on the exterior side, producing a single outward protrusion. In the most common construction, the triangle is isosceles with equal sides of length l = sqrt((w/2)^2 + t^2).

The area of a rectangleja is the sum of the areas of the rectangle and the attached triangle: A = w h + 1/2 w t. The perimeter depends on the triangle's shape. For the isosceles case, with the two equal sides length l, the boundary consists of the bottom edge w, the two vertical sides h, and the two triangle sides, giving P = w + 2 h + 2 l. In the general case, where the two triangle sides adjacent to the base have lengths a and b, the perimeter is P = w + 2 h + a + b.

Variants of the construction include attaching the triangle to any side of the rectangle or using a non-isosceles triangle, resulting in a pentagon with unequal triangle sides or a different orientation. The rectangleja remains convex when the triangle protrudes outward from the rectangle; it can become concave if the triangle is drawn inward.

Rectangleja is used primarily as a teaching or design construct to illustrate area composition, polygonal properties, and the effects of attaching shapes to a base. Related shapes include the pentagon, rectangle, and triangle.

See also: pentagon, rectangle, triangle, irregular polygon.