Do you know the house of squares? There are many different quadrilaterals, including the trapezoid. What a trapezoid is and which trapezoids there are, you will find out in this explanation.
Trapezoid – definition
Every rectangle or square is a trapezium, but not every trapezium is a rectangle or square. The basic requirement for a trapezoid is that it has two parallel sides. These have the square and the rectangle, so they are trapezoids.
A trapeze is a special quadrilateral where at least two opposite sides are parallel to each other. The parallel sides of a trapezoid are called the bases and the other two sides are the legs.
The following applies:
\
The longer side of the two basic sides is also called the base.
A trapeze possesses in addition to the four Corners \(A,\,B,\,C,\,D\) and Pages \(a,\,b,\,c,\,d\)four Angle \(\alpha,\,\beta,\,\gamma,\,\delta\)two diagonal \(e,\,f\)one Height \(h\) and a Mean parallel \(m\).
Figure 1: Trapezoid definition
Trapezoid properties
In order to analyze a trapezoid, you need to know some properties of the trapezoid.
the sum of interior angles of a trapezium is 360° and is calculated from the added angles of the trapezium.
The following applies:
\
The angles of a trapezoid can be of different sizes as long as they add up to 360°. The diagonals of a trapezium also have special features. While the two diagonals of other quadrilaterals, such as the parallelogram, are the same length, you have to calculate both of the diagonals of a trapezoid.
The two diagonals of a trapezoid e and f can be calculated using the following formula:
\begin {align} e&=\sqrt {\frac {{\color{#1478c8}a}\cdot {\color{#8363e2}d}^{2}+{\color{#1478c8}a}^{2 }\cdot {\color{#fa3273}c}-{\color{#1478c8}a}\cdot {\color{#fa3273}c}^{2}-{\color{#fa3273}c}\cdot { \color{#00dcb4}b}^{2}}{{\color{#1478c8}a}-{\color{#fa3273}c}}}\hspace{0.5cm} (\text{for}\, a \neq c) \\f&=\sqrt {\frac {{\color{#1478c8}a}\cdot {\color{#00dcb4}b}^{2}+{\color{#1478c8}a}^{ 2}\cdot {\color{#fa3273}c}-{\color{#1478c8}a}\cdot {\color{#fa3273}c}^{2}-{\color{#fa3273}c}\cdot {\color{#8363e2}d}^{2}}{{\color{#1478c8}a}-{\color{#fa3273}c}}}\hspace{0.5cm} (\text{for}\, a\neq c) \end{align}
Figure 2: Trapezoidal properties diagonals
Before you start calculating the diagonals, you should always check which of the sides are parallel and therefore the base sides. In addition, you must then check whether the base sides are of equal length.
You can find out how to calculate the diagonal when the base sides are the same length in the explanation “Diagonal Rectangle”.
Calculate trapezoid area
Every geometric figure also has an area.
The area of a trapezium is calculated by dividing the height by half and the sum of the sides a and c, if \(a\parallel c\) applies.
The following then applies:
\
You can also calculate the area of a trapezoid using the product of the median line m and the height h.
The explanation «Area of trapezium» explains how to use and derive this formula.
To calculate the area of a trapezoid, you need the height of the trapezium. You can now find out how to calculate this.
Calculate trapezoid height
In order to calculate the area of the trapezoid, you need the height of the trapezium. This can be given in the task or you have to calculate it first.
You can calculate the height of a trapezoid using different formulas.
- You can use the Heron formula to calculate the height.\begin{align} h&=\frac {2}{|ac|}\cdot \sqrt {s\cdot (s-|ac|)\cdot (sb)\cdot ( sd)}\hspace{0.5cm} (\text{for}\, a\neq c) \\ \text{with}\, s&=\frac {|ac|+b+d}{2}\end{ align}
- You can change the formula for the area to h and get \
- You can also calculate the height of a trapezoid using the product of a leg and one of the adjacent angles. \
The formula you use to calculate the height depends on the task and the given values. If you are supposed to calculate the area and first need the height, it is not effective to use the formula with the area.
exercise 2
Calculate the height of the trapezium ABCD with the side lengths \(a=8\,cm,\, b=5\,cm,\, c=5\,cm\,\,\text{and}\,\,d= 4\,cm\) and right angles on the leg d.
example
You can calculate the height here in two ways.
1st variant: heroic formula
At the beginning you calculate the s with the formula \(s=\frac {|ac|+b+d}{2}\).
\begin{align} s&=\frac {|ac|+b+d}{2}\\ s&=\frac {|8-5|+5+4}{2} \\ s&=\frac{12} {2}\\ s&=6 \end{align}
Then you put the s and all other values into the formula for calculating the height h.
\begin{align} h&=\frac {2}{|ac|}\cdot \sqrt {s\cdot (s-|ac|)\cdot (sb)\cdot (sd)} \\ h&=\frac { 2}{|8-5|}\cdot \sqrt {6\cdot (6-|8-5|)\cdot (6-5)\cdot (6-4)} \\ h&=\frac {2} {3}\cdot \sqrt {6\cdot 3\cdot 1 \cdot 2} \\ h&=\frac{2}{3}\cdot\sqrt{36} \\ h&=4\,\end{align}
2nd variant: Angle
First you look for the leg for which you have given a fitting angle. In this exercise you are given a right angle on the leg d. Then you put these values into the formula and calculate the height h.
\begin{align} h&=d\cdot \sin(\delta )=d\cdot \sin(\alpha ) \\ h&=4\cdot \sin(90°)=4\cdot \sin{(90°) } \\ h&=4\, \end{align}
The height of the trapezium is 4 cm.