Isosceles Trapezoid Calculator

What is an Isosceles Trapezoid?

An isosceles trapezoid (or isosceles trapezium in British English) is a convex quadrilateral with at least one pair of parallel sides (called bases $a$ and $b$) and non-parallel lateral sides (called legs $c$ and $d$) of equal length ($c = d$).

Due to its bilateral reflection symmetry along the perpendicular bisector of its bases, an isosceles trapezoid possesses remarkable geometric properties:

The base angles adjacent to base $a$ are equal ($\alpha = \beta$).
The base angles adjacent to base $b$ are equal ($\gamma = \delta = 180^\circ - \alpha$).
Opposite angles are supplementary ($\alpha + \gamma = 180^\circ$), meaning every isosceles trapezoid is a cyclic quadrilateral that can be inscribed in a unique circumscribed circle.
Both diagonals have identical lengths ($D_1 = D_2 = D$).

Isosceles Trapezoid Formulas

Let $a$ and $b$ be the lengths of parallel bases, $c$ be the leg length, $h$ be the perpendicular height, and $\Delta = \frac{|a - b|}{2}$ be the horizontal projection:

Height ($h$): $h = \sqrt{c^2 - \left(\frac{a - b}{2}\right)^2} = \sqrt{c^2 - \Delta^2}$
Leg ($c$): $c = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2} = \sqrt{h^2 + \Delta^2}$
Midsegment ($m$): $m = \frac{a + b}{2}$
Area ($A$): $A = \frac{a + b}{2} \times h = m \times h$
Perimeter ($P$): $P = a + b + 2c$
Diagonals ($D$): $D = \sqrt{a \cdot b + c^2} = \sqrt{h^2 + \left(\frac{a + b}{2}\right)^2}$
Circumcircle Radius ($R$): $R = \frac{c \cdot D}{2h}$
Base Angle ($\alpha$): $\alpha = \arccos\left(\frac{\Delta}{c}\right) = \arctan\left(\frac{h}{\Delta}\right)$

How to Calculate an Isosceles Trapezoid

Follow these steps depending on the known measurements:

From Two Bases ($a, b$) and Leg ($c$): First find the horizontal offset $\Delta = \frac{|a - b|}{2}$. The height is $h = \sqrt{c^2 - \Delta^2}$. Then compute area $A = \frac{a+b}{2} \times h$ and perimeter $P = a + b + 2c$.
From Two Bases ($a, b$) and Height ($h$): Find $\Delta = \frac{|a - b|}{2}$. Calculate leg length $c = \sqrt{h^2 + \Delta^2}$, base angle $\alpha = \arctan(h / \Delta)$, and diagonal $D = \sqrt{ab + c^2}$.
From Two Bases ($a, b$) and Base Angle ($\alpha$): Compute $h = \Delta \times \tan(\alpha)$ and leg $c = \frac{\Delta}{\cos(\alpha)}$.

Related Geometry Calculators

Explore other polygon and quadrilateral calculation tools:

Area of a Trapezoid Calculator: Quick general formula for any trapezoid given bases and height.
Irregular Trapezoid Area Calculator: Solve non-isosceles trapezoids with four unequal sides.
Isosceles Triangle Calculator: Calculate sides, apex angles, and area of isosceles triangles.
Isosceles Right Triangle Calculator: 45-45-90 special right triangle calculator.

Frequently Asked Questions

What makes a trapezoid an isosceles trapezoid?

A trapezoid is isosceles if its two non-parallel lateral sides (legs) are equal in length. This symmetry guarantees equal base angles and equal diagonals.

Can an isosceles trapezoid be a rectangle?

Yes. If the base angles of an isosceles trapezoid are $90^\circ$, then $a = b$ and the shape becomes a rectangle (or a square if the height equals the base).

Can an isosceles trapezoid have an incircle (inscribed circle)?

An isosceles trapezoid has an incircle if and only if the sum of its bases equals the sum of its legs ($a + b = 2c$), which makes it a tangential quadrilateral with inradius $r = \frac{h}{2} = \frac{\sqrt{ab}}{2}$.

Why are the diagonals of an isosceles trapezoid equal?

By constructing right triangles from the vertices to the opposite base, each diagonal forms the hypotenuse of a right triangle with legs $h$ and $a - \Delta = \frac{a + b}{2}$. Since both triangles have identical legs, the diagonals must be equal: $D = \sqrt{h^2 + \left(\frac{a+b}{2}\right)^2}$.

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