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Properties of a Circular area
Properties of an Elliptical Area
Properties of a Semi-Circular area
Properties of a Circular Sector
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Moment of Inertia of Trapezoid

Properties of a Trapezoid

- By Dr. Minas E. Lemonis, PhD - Updated: April 14, 2019

This tool calculates the basic geometric properties of a trapezoid. Enter below the shape dimensions. It is not required for base α to be the bigger one. The calculated results will have the same units as your input. Please use consistent units for any input.

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b =
h =
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Geometric properties:
Area =
Perimeter =
xc =
yc =
Lengths:
Side c =
Side d =
Diagonal p =
Diagonal q =
Angles :
φ1 =
φ2 =
φ3 =
φ4 =
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Definitions

Geometry

Trapezoid is a quadrilateral shape with at least two parallel sides. The definitions shown in the following figure are used:

shape geometry

The area of a trapezoid is given by the formula:

A = h\frac{a+b}{2}

where a, b the lengths of the two bases and h the height.

The perimeter of a trapezoid is simply the sum of the lengths of all sides:

P = a+b+c+d

Finding the lengths of the non parallel sides c and d, can be done if one interior angle of the trapezoid is known. Let's assume that angle φ1is given. Using simple geometrical principles, for the right triangles, with sides c, d as hypotenuses (see figure below), the calculation of lengths c and d can by done:

\begin{split} & c & = \frac{h}{\sin{\varphi_1}} \\ & \alpha_1 & = \sqrt{c^2 - \alpha_1^2} \\ & \alpha_2 & = a - b - \alpha_1\\ & d & = \sqrt{\alpha_2^2 + h^2} \end{split}

shape geometry

Interior angle φ3is supplementary with φ1, since bases a and b are parallel. Therefore:

\varphi_3 =180^{\circ} -\varphi_1

The remaining interior angles (which are also supplementary) can be found using simple geometrical principles, for the right triangle, with side d as hypotenuse:

\begin{split} & \tan{\varphi_2} & = \frac{h}{\alpha_2} \Rightarrow \varphi_2 = \tan^{-1}\left(\frac{h}{\alpha_2}\right)\\ & \varphi_4 & = 180^{\circ} - \varphi_2 \end{split}

There are many ways to find the lengths of diagonals, once the sides or the interior angles are known. Here, a solution employing the Pythagorean Theorem on the highlighted right triangle (see next figure) is presented, for diagonal p:

p = \sqrt{h^2 + \left(\alpha-\alpha_2\right)^2}

Similarly, the other diagonal is found as:

q = \sqrt{h^2 + \left(\alpha-\alpha_1\right)^2}

shape geometry

Centroid

The centroid coordinates in respect to the bottom base left vertex, xcand yc(see figure below) can be calculated using the first moments of area, of the three sub-areas A,B,C.

shape geometry

For xc, considering the first moments of area, relative to the part B middle, it is found:

\begin{split} & A\left(x_{c}-a_1-\frac{b}{2}\right) = \frac{a_1 h}{2}\left(-\frac{b}{2}-\frac{a_1}{3}\right) + \frac{a_2 h}{2}\left(\frac{b}{2}+\frac{a_2}{3}\right) \Rightarrow \\ \\ & x_{c} = a_1 +\frac{b}{2} + \frac{h\left(a_2-a_1\right)\left(\frac{3}{2}b+a_1+a_2 \right)}{6A} \end{split}

where A is the area of the trapezoid.

As for the yc, considering the first moments of area relative to the bottom base, it is found:

\begin{split} & A y_{c} = \frac{a_1 h}{2}\frac{h}{3} + b h\frac{h}{2} + \frac{a_2 h}{2}\frac{h}{3} \Rightarrow \\ \\ & y_{c} = \frac{h^2\left(3b + a_1+a_2\right)}{6A} = \frac{h^2\left(2b + a\right)}{6A} \\ \\ \end{split}

The above formulas are valid even when α1or α2are negative, which occurs when the angles φ1or φ2, are obtuse.

See also
Properties of a Circular area
Properties of an Elliptical Area
Properties of a Semi-Circular area
Properties of a Circular Sector
All Geometric Shapes tools
Moment of Inertia of Trapezoid