Isosceles Triangle


An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length b and the remaining side has length a. This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos (leg).

A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal. Another special case of an isosceles triangle is the isosceles right triangle.

The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as


The area is therefore given by


The inradius of an isosceles triangle is given by


The mean of y is given by


so the geometric centroid is


or 2/3 the way from its vertex (Gearhart and Schulz 1990).


Considering the angle at the apex of the triangle and writing R instead of b, there is a surprisingly simple relationship between the area and vertex angle theta. As shown in the above diagram, simple trigonometry gives


so the area is


Erecting similar isosceles triangles on the edges of an initial triangle DeltaABC gives another triangle DeltaA^'B^'C^' such that AA^', BB^', and CC^' concur. The triangles are therefore perspective triangles.

No set of n>6 points in the plane can determine only isosceles triangles.

See also

30-60-90 Triangle, Acute Triangle, Equilateral Triangle, Golden Gnomon, Golden Triangle, Isosceles Right Triangle, Isosceles Tetrahedron, Isoscelizer, Kiepert Parabola, Obtuse Triangle, Petr-Neumann-Douglas Theorem, Point Picking, Pons Asinorum, Right Triangle, Scalene Triangle, Steiner-Lehmus Theorem Explore this topic in the MathWorld classroom

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Gearhart, W. B. and Schulz, H. S. "The Function sinx/x." College Math. J. 21, 90-99, 1990.

Cite this as:

Weisstein, Eric W. "Isosceles Triangle." From MathWorld--A Wolfram Web Resource.

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