You may come across Trapezoids when you see a wheelbarrow in a garden or when you pass over a bridge and take a look at its trusses. These geometric shapes are important in applications of architecture and construction. You may already be familiar with how to calculate the area of Triangles, which will be useful for this article when we take a look at the trapezoid's area formula and some examples of how to use it.
Let's start by recalling what a trapezoid is.
Trapezoid Definition
A trapezoid is a quadrilateral (four-sided plane figure), that has exactly one pair of parallel sides.
The following figure is a trapezoid.
Fig. 1. Trapezoid illustration.
In the above figure, the parallel sides (in this case, \(\overline{AD}\) and \(\overline{BC}\)) are referred to as bases of the trapezoid. The non-parallel sides (\(\overline{AB}\) and \(\overline{DC}\)) are referred to as the legs of the trapezoid.
A trapezoid is also commonly referred to as a trapezium.
Definition of the Area of a Trapezoid
The area of a trapezoid is defined by the space enclosed within its boundaries, as occupied in a two-dimensional plane.
The area of a trapezoid is measured in square units such as \(\text{m}^2\), \(\text{cm}^2\), \(\text{in}^2\), \(\text{ft}^2\), etc.
Formula for the Area of a Trapezoid
Consider the following trapezoid:
Fig. 2. Trapezoid with bases, \(a\) and \(b\), and height \(h\).
The area of a trapezoid is given by the formula:
\[\text{Area} = \frac{1}{2} h (a + b)\]
where:
\(h \Rightarrow\) height of the trapezoid (perpendicular distance between the bases),
\(a, b \Rightarrow\) lengths of the bases.
How did we get this formula, you ask? Let's show you.
Recall that the area of a triangle is given by the formula:
\[\text{Area} = \frac{1}{2} \text{base} \cdot \text{height}\]
We can divide the above trapezoid into two triangles along either of the diagonals. Let us take the diagonal \(\overline{BD}\), and divide the trapezoid into the triangles \(\triangle{BAD}\) and \(\triangle{BCD}\).
Fig. 3. Trapezoid divided into \(\triangle{BAD}\) and \(\triangle{BCD}\) by the diagonal \(\overline{BD}\).
Then we can say that,
\[\begin{align}\text{Area of trapezoid ABCD} & = \text{Area of } \triangle{BAD} + \text{Area of } \triangle{BCD} \\ \\& = \frac{1}{2} b \cdot h + \frac{1}{2} a \cdot h \\ \\& = \frac{1}{2} h (a + b)\end{align}\]
Think of a parallelogram, where both pairs of opposite sides are parallel. You can apply the above formula to derive the formula for the area of a parallelogram too.
\[\begin{align}\text{Area} & = \frac{1}{2} h (a + b) \\ \\& = \frac{1}{2} h (b + b) \qquad \text{Opposite sides of a parallelogram are of equal length} \\ \\& = \frac{1}{2} h (2b) \\ \\& = b \cdot h\end{align}\]
This is the formula for the area of a parallelogram.
Examples of the Area of a Trapezoid
Now let us look at some examples related to the area of Trapezoids.
A trapezoid has bases of lengths \(10\,\text{cm}\) and \(15\,\text{cm}\). The perpendicular distance between the bases is \(8\,\text{cm}\). Find the area of the trapezoid.
Solution
To solve this problem, we simply need to substitute the values of the lengths of the bases and the height in the area of trapezoid formula.
\[\begin{align}\text{Area} & = \frac{1}{2} h (a + b) \\ \\& = \frac{1}{2} \cdot 8 (10 + 15) \\ \\& = 4 \cdot 25 \\ \\& = 100\,\text{cm}^2\end{align}\] The area of the trapezoid is \( 100\,\text{cm}^2 \).
Now let's see an example using the coordinate plane.
Find the area of the following trapezoid.
Fig. 4. Trapezoid on the coordinate plane.
Solution
In this case, to be able to find the area of the above trapezoid, we need to find the length of the bases and the height of the trapezoid.
These values are not given, but we can use the coordinate plane to work them out.
We need to calculate the distance between each of the points, how can we do this?
The distance between the points \(B(6, 2)\) and \(C(9, 2)\) can be calculated by finding the absolute value of the difference between their x-coordinates, using \(|x_2 - x_1|\). The same applies for the distance between the points \(A(2, 7)\) and \(D(10, 7)\).
The distance between the points \(B(6, 2)\) and \(E(6, 7)\) can be calculated by finding the absolute value of the difference between their y-coordinates, using \(|y_2 - y_1|\).
\[\begin{align}a &= \overline{BC} = |x_2 - x_1| = |9 - 6| = 3 \\ \\b &= \overline{AD} = |x_2 - x_1| = |10 - 2| = 8 \\ \\h &= \overline{BE} = |y_2 - y_1| = |7 - 2| = 5 \end{align}\]Now that we have all the values that we need, we can substitute them in the area of trapezoid formula.
\[\begin{align}\text{Area} & = \frac{1}{2} \cdot 5 (3 + 8) \\ \\& = \frac{5}{2} \cdot (11) \\ \\& = \frac{55}{2} \\ \\& = 27.5\,\text{units}^2\end{align}\] The area of the trapezoid is \( 27.5\,\text{units}^2 \).
A trapezoid with an area of \(35\,\text{m}^2\) has bases of lengths, \(3\,\text{m}\) and \(4\,\text{m}\). Find the distance between the parallel sides.
Solution
The distance between the parallel sides is the height of the trapezoid. So, let's substitute the values that we have in the area of trapezoid formula, and then solve for \(h\).
\[\begin{align}\text{Area} & = \frac{1}{2} h (a + b) \\ \\35 & = \frac{1}{2} \cdot h (3 + 4) \\ \\35 & = \frac{7 \cdot h}{2} \\ \\h & = \frac{35 \cdot 2}{7} \\ \\& = \frac{70}{7} \\ \\& = 10\,\text{m}\end{align}\]
The height of the trapezoid is \(10\, \text{m}\).
Area of Trapezoid Without Known Height
If you are given a trapezoid with the lengths of all its bases and legs, but no height is provided, then you need to calculate its height first, to be able to find the area of the trapezoid. Let's see an example to show you what to do in this case.
Find the area of the following trapezoid.
Fig. 5. Trapezoid with no height example.
Notice that the legs of the trapezoid are of equal lenght \(6\, \text{m}\), therefore, this is an isosceles trapezoid, and we can calculate its height as follows.
Fig. 6. Height of a trapezoid using Pythagoras.
Notice that we have a right triangle at each side. The bases of each triangle was calculated by finding the difference between \(18\) and \(10\), and then dividing the result by \(2\).
\[18 - 10 = \frac{8}{2} = 4\, \text{m}\]
Now we can calculate the height using the Pythagoras Theorem which states that, in a right triangle, the square of the hypothenuse equals the sum of the other two sides squared.
\[\begin{align}c^2 & = a^2 + b^2 \\ \\6^2 & = 4^2 + h^2 \\ \\36 & = 16 + h^2 \\ \\h^2 & = 36 - 16 \\ \\h & = \sqrt{20}\\ \\& = 4.47\, \text{m}\end{align}\]Now that we know the length of the height, we can calculate the area of the trapezoid.
\[\begin{align}\text{Area} & = \frac{1}{2} \cdot 4.47 (10 + 18) \\ \\& = \frac{1}{2} \cdot 4.47 \cdot 28 \\ \\& = 62.6\, \text{m}^2\end{align}\] The area of the trapezoid is \(62.6\, \text{m}^2\).
Area of a Trapezoid When Given Diagonals
Another interesting scenario is when you need to calculate the area of a trapezoid when only the lengths of its diagonals and the angle between them are given.
Consider a trapezoid with diagonals of lengths, \(d_1\) and \(d_2\), and an angle of \(\alpha\) between them.
Fig. 7. Trapezoid with diagonals \(d_1\) and \(d_2\).
In this case, the area of the trapezoid is given by
\[\text{Area} = \frac{1}{2} d_1 \cdot d_2 \cdot \sin(\alpha)\]
The length of the diagonals of a trapezoid \(d_1\) and \(d_2\) is \(6.4\, \text{m}\), and the angle \(\alpha\) between them measures \(77.3^{\circ}\). Find the area of the trapezoid.
\[\text{Area} = \frac{1}{2} \cdot 6.4 \cdot 6.4 \cdot \sin(77.3^{\circ}) = 19.98\, \text{m}^2\]
The area of the trapezoid is \(19.98\, \text{m}^2\).
Area of trapezoids - Key takeaways
- A trapezoid is a quadrilateral that has exactly one pair of parallel sides.
- The area of a trapezoid is defined by the space enclosed within its boundaries, as occupied in a two-dimensional plane.
- The area of a trapezoid is given by the formula: \(\text{Area} = \frac{1}{2} h (a + b)\).
- If you are given a trapezoid with the lengths of all its bases and legs, but no height is provided, you need to calculate its height first, using the Pythagoras Theorem, to be able to find the area of the trapezoid.
- When you need to calculate the area of a trapezoid when only the lengths of its diagonals and the angle between them are given, you can use the formula: \(\text{Area} = \frac{1}{2} d_1 \cdot d_2 \cdot \sin(\alpha)\).
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