Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.

The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base.

Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base.

The two equal angles at the base (opposite the legs) are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.

[4] "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg).

[9] Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex.

[14] For any isosceles triangle, the following six line segments coincide: Their common length is the height

However, applying Heron's formula directly can be numerically unstable for isosceles triangles with very sharp angles, because of the near-cancellation between the semiperimeter and side length in those triangles.

from one of the two equal-angled vertices satisfies[27] as well as and conversely, if the latter condition holds, an isosceles triangle parametrized by

[28] The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles.

The 30-30-120 isosceles triangle makes a boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external).

[31] The radius of the inscribed circle of an isosceles triangle with side length

is:[17] The center of the circle lies on the symmetry axis of the triangle, this distance above the base.

[11] A much older theorem, preserved in the works of Hero of Alexandria, states that, for an isosceles triangle with base

, the side length of the inscribed square on the base of the triangle is[33] For any integer

[31] Generalizing the partition of an acute triangle, any cyclic polygon that contains the center of its circumscribed circle can be partitioned into isosceles triangles by the radii of this circle through its vertices.

The fact that all radii of a circle have equal length implies that all of these triangles are isosceles.

[38] Isosceles triangles commonly appear in architecture as the shapes of gables and pediments.

[39] The Egyptian isosceles triangle was brought back into use in modern architecture by Dutch architect Hendrik Petrus Berlage.

[40] Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength.

[41] Surfaces tessellated by obtuse isosceles triangles can be used to form deployable structures that have two stable states: an unfolded state in which the surface expands to a cylindrical column, and a folded state in which it folds into a more compact prism shape that can be more easily transported.

[42] The same tessellation pattern forms the basis of Yoshimura buckling, a pattern formed when cylindrical surfaces are axially compressed,[43] and of the Schwarz lantern, an example used in mathematics to show that the area of a smooth surface cannot always be accurately approximated by polyhedra converging to the surface.

[44] In graphic design and the decorative arts, isosceles triangles have been a frequent design element in cultures around the world from at least the Early Neolithic[45] to modern times.

[47] They also have been used in designs with religious or mystic significance, for instance in the Sri Yantra of Hindu meditational practice.

[49] In celestial mechanics, the three-body problem has been studied in the special case that the three bodies form an isosceles triangle, because assuming that the bodies are arranged in this way reduces the number of degrees of freedom of the system without reducing it to the solved Lagrangian point case when the bodies form an equilateral triangle.

[50] Long before isosceles triangles were studied by the ancient Greek mathematicians, the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area.

[51] The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid.

[52] This result has been called the pons asinorum (the bridge of asses) or the isosceles triangle theorem.

[53] A well-known fallacy is the false proof of the statement that all triangles are isosceles, first published by W. W. Rouse Ball in 1892,[54] and later republished in Lewis Carroll's posthumous Lewis Carroll Picture Book.

[55] The fallacy is rooted in Euclid's lack of recognition of the concept of betweenness and the resulting ambiguity of inside versus outside of figures.