Isosceles trapezoid

Characterizations

If a quadrilateral is known to be a trapezoid, it is not sufficient just to check that the legs have the same length in order to know that it is an isosceles trapezoid, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides.

Any one of the following properties distinguishes an isosceles trapezoid from other trapezoids:

• The diagonals have the same length.
• The base angles have the same measure.
• The segment that joins the midpoints of the parallel sides is perpendicular to them.
• Opposite angles are supplementary, which in turn implies that isosceles trapezoids are cyclic quadrilaterals.
• The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, ${\displaystyle AE=DE}$, ${\displaystyle BE=CE}$, and ${\displaystyle AE\ne CE}$ if one wishes to exclude rectangles).

Formula

In this article, let ${\displaystyle ABCD}$ be an isosceles trapezoid. The edges ${\displaystyle AD}$ and ${\displaystyle BC}$ are the bases of, ${\displaystyle AC}$ and ${\displaystyle BD}$ are the diagonals of, and ${\displaystyle E}$ is the intersection between two diagonals of an isosceles trapezoid.

Angles

In an isosceles trapezoid, the base angles have the same measure pairwise. In the picture below, angles ${\displaystyle \mathrm{\angle }ABC}$ and ${\displaystyle \mathrm{\angle }DCB}$ are obtuse angles of the same measure, while angles ${\displaystyle \mathrm{\angle }BAD}$ and ${\displaystyle \mathrm{\angle }CDA}$ are acute angles, also of the same measure. Since the lines ${\displaystyle AD}$ and ${\displaystyle BC}$ are parallel, angles adjacent to opposite bases are supplementary. That is: $${\displaystyle \mathrm{\angle }ABC+\mathrm{\angle }BAD={180}^{\circ }.}$$

Diagonals and height

The diagonals of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an equidiagonal quadrilateral. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals ${\displaystyle AC}$ and ${\displaystyle BD}$ have the same length ${\displaystyle AC=BD}$ and divide each other into segments of the same length ${\displaystyle AE=DE}$ and ${\displaystyle BE=CE}$.

The ratio in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is, $${\displaystyle \frac{AE}{EC}=\frac{DE}{EB}=\frac{AD}{BC}.}$$

The length of each diagonal is, according to Ptolemy's theorem, given by $${\displaystyle p=\sqrt{ab+c^2}}$$ where ${\displaystyle a}$ and ${\displaystyle b}$ are the lengths of the parallel sides ${\displaystyle AD}$ and ${\displaystyle BC}$, and ${\displaystyle c}$ is the length of each leg ${\displaystyle AB}$ and ${\displaystyle CD}$.

The height is, according to the Pythagorean theorem, given by $${\displaystyle h=\sqrt{p^2-{\left(\frac{a+b}{2}\right)}^2}=\frac{1}{2}\sqrt{4c^2-(a-b{)}^2}.}$$

The distance from point ${\displaystyle E}$ to base ${\displaystyle AD}$ is given by $${\displaystyle d=\frac{ah}{a+b}}$$ where ${\displaystyle a}$ and ${\displaystyle b}$ are the lengths of the parallel sides ${\displaystyle AD}$ and ${\displaystyle BC}$, and ${\displaystyle h}$ is the height of the trapezoid.

Area

The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (the parallel sides) times the height. If the edges ${\displaystyle AD=a}$ and ${\displaystyle BC=b}$, and the height ${\displaystyle h}$ is the length of a line segment between ${\displaystyle AD}$ and ${\displaystyle BC}$ that is perpendicular to them, then the area ${\displaystyle K}$ is $${\displaystyle K=\frac{1}{2}\left(a+b\right)h.}$$

If instead of the height of the trapezoid, the common length of the legs ${\displaystyle AB=CD=c}$ is known, then the area can be computed using Brahmagupta's formula for the area of a cyclic quadrilateral, which, with two sides equal, simplifies to $${\displaystyle K=(s-c)\sqrt{(s-a)(s-b)},}$$ where ${\displaystyle s=\frac{1}{2}(a+b+2c)}$ is the semi-perimeter of the trapezoid. This formula is analogous to Heron's formula to compute the area of a triangle. The previous formula for area can also be written as $${\displaystyle K=\frac{a+b}{4}\sqrt{(a-b+2c)(b-a+2c)}.}$$

Circumradius

The radius in the circumscribed circle is given by $${\displaystyle R=c\sqrt{\frac{ab+c^2}{4c^2-(a-b{)}^2}}.}$$

In a rectangle where ${\displaystyle a=b}$, this is simplified to ${\displaystyle R=\frac{1}{2}\sqrt{a^2+c^2}}$.

Duality

Kites and isosceles trapezoids are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid. For any isosceles trapezoid, tangent lines to the circumscribing circle at its four vertices form the four sides of a kite. This correspondence can also be seen as an example of polar reciprocation, a general method for corresponding points with lines and vice versa given a fixed circle. Although they do not touch the circle, the four vertices of the kite are reciprocal in this sense to the four sides of the isosceles trapezoid. The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.