Geometry and Locus: Loci of Geometrical Figures, Shapes

May 20, 202239 Insightful Publication

A **trapezium **is a quadrilateral with one set of parallel sides (bases) and non-parallel sides (legs). The number of unit squares that can be fit into an **Area of Trapezoid** is measured in square units as \(cm2, m2, in2\), etc.

Imagine a trapezium ABCDABCD such that ABAB and CDCD are its two parallel sides while ADAD and BCBC are its non-parallel sides. The distance between the two parallel sides is called the height or altitude of the trapezoid.

In this article, we will study the formula for the area of a trapezoid using the base and height of a trapezium and using two triangles formed when we join the two corners of a trapezium.

A trapezoid or trapezium is a quadrilateral, defined as a shape with four sides and one set of parallel sides. The parallel sides are called the bases, while the non-parallel sides are called the legs of the trapezoid.

The line connecting the midpoints of the non-parallel sides of a trapezium is called the mid-segment.

**Learn All the Concepts on Area of Trapezium**

Check above the different types of trapezium images, where the arrow represents the parallel side of it. In all three images, we can see, the two sides are parallel to each other, whereas the other two sides are non-parallel

If we draw a line segment between the two non-parallel sides, the trapezium will be divided into two unequal parts.

We have learned the concept of isosceles triangles, where the two sides of a triangle are equal, and the angles opposite to the equal sides are also equal. In the same way, we have an isosceles trapezium, where the two non-parallel sides are equal and form equal angles at one of the bases. We can see the example of it in the below figure given. It is also called an isosceles trapezium.

A right trapezoid (also called a right-angled trapezoid) has two adjoining right angles. Right trapezoids are used in the trapezoidal rule for calculating areas under a curve. An acute trapezium has two adjoining acute angles on its longer base edge, while an obtuse trapezoid has one acute and one obtuse angle on each base.

Below are some properties of the trapezoid.

- Like other quadrilaterals, the sum of the interior angles of the trapezium is equal to \({360^ \circ }\).
- A trapezium has two parallel and non-parallel sides.
- The diagonals of isosceles trapezium divide each other.
- The length of the mid-segment is equal to half the total of the parallel sides in a trapezium.
- Two pairs of adjacent angles of a trapezium formed between the parallel sides, and one of the non-parallel sides are supplementary.

The area of a trapezoid can be computed using the lengths of two of its parallel sides and the height between them. The formula to calculate the area \((A)\) of a trapezium using base and height is given as,

\(A = \frac{1}{2}(a + b) \times h\)

where \(a\) and \(b\) are the lengths of the parallel sides of the trapezoid, and \(h\) is the height.

Let \(ABCD\) be the trapezium such that \(AB\) and \(CD\) are its two parallel sides while \(AD\) and \(BC\) are its non-parallel sides.

Let \(DL=h\) be the height of the trapezium \(ABCD. AC\) divided trapezium into two triangles \(ABC\) and \(ACD\).

Therefore, Area of trapezium \(ABCD = {\text{Area}}\,{\text{of}}\,\Delta ABC + {\text{Area}}\,{\text{of}}\,ACD\) …(I)

Since \(h\) is the altitude of trapezium \(ABCD\). Therefore, it is also the altitude of \(\Delta ABC\) and \(\Delta ACD\).

Therefore, the area of \(\Delta ABC = \frac{1}{2} \times AB \times h\)

and, the area of \(\Delta ACD = \frac{1}{2} \times DC \times h\)

Substituting these values in (i), we get

Area of trapezium \(ABCD = \left( {\frac{1}{2} \times AB \times h} \right) + \left( {\frac{1}{2} \times DC \times h} \right)\)

\( = \frac{1}{2} \times (AB + DC) \times h\)

\(= \frac{1}{2} \times \left( {{\rm{Sum\;of\;parallel\;sides}}} \right) \times \left( {{\rm{distance\;between\;parallel\;sides}}} \right)\)

Hence, the area of a trapezium equals half the sum of parallel sides multiplied by the altitude.

To derive the formula for the area of a trapezium using parallelogram, we will consider two identical trapezoids, each with bases \(a\) and \(b\) and height \(h\). Let \(A\) be the area of each trapezoid. Imagine that the second trapezium is turned upside down, as shown in the figure below.

Joining the two trapezoids, we get

We can see that the new figure obtained by attaching the two trapeziums is a parallelogram whose base is \(a+ b\) and whose height is \(h\). We know that

The area of a parallelogram \( = {\rm{ base }} \times {\rm{ height }}\)

The area of the above parallelogram is, \(A + A = 2A\).

Therefore, \(2{\rm{ }}A = (a + b){\rm{ }}h\)

\(\Rightarrow A = \frac{1}{2}(a + b) \times h\)

Thus, we got the area of the trapezium.

A three-dimensional solid made up of two trapezoids on opposite faces joined by four parallelograms called the lateral faces is known as a trapezoidal prism.

If \(a,b\) are the base sides, \(h\) is the height and \(l\) be the length. Then the volume of the trapezoidal prism is given by

\(V = \frac{1}{2}(a + b) \times h \times l\)

The surface area of the trapezoidal prism \( = 2 \times {\rm{area\;of\;base}} + {\rm{lateral\;surface\;area}}\)

\( = 2 \times \frac{1}{2}(a + b)h + (a \times l) + (b \times l) + (c \times l) + (d \times l)\)

\( = h(a + b) + l(a + b + c + d)\)

Therefore, the total surface area of the trapezoidal prism is \(h(a + b) + l(a + b + c + d)\)

** Q.1. Find the area of a trapezium whose parallel sides are of lengths \({\rm{10\,cm}}\) and \({\rm{12\,cm}}\), and the distance between them is \({\rm{4\,cm}}\)** We know that, area of a trapezium

Ans:

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

\( = \frac{1}{2} \times (10 + 12) \times 4\;{\rm{c}}{{\rm{m}}^2}\)

\( = \frac{1}{2} \times 22 \times 4\;{\rm{c}}{{\rm{m}}^2}\)

\( = 11 \times 4\;{\rm{c}}{{\rm{m}}^2}\)

\( = 44\,{\text{c}}{{\text{m}}^2}\)

Therefore, the area of the given trapezium is \( = 44\,{\text{c}}{{\text{m}}^2}\)

** Q.2. The area of a trapezium is \(440\;{\rm{c}}{{\rm{m}}^2}\). The lengths of the parallel sides are respectively \(30\;{\rm{cm}}\) and \(14\;{\rm{cm}}\). Find the distance between them.** Let the distance between the parallel sides be \(h\,{\rm{cm}}.\) Also, area \( = 440\;{\rm{c}}{{\rm{m}}^2}\) and the length of the parallel sides is \(30\;{\rm{cm}}\) and \(14\;{\rm{cm}}\), respectively.

Ans:

We know that, area of a trapezium

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

\( \Rightarrow 440 = \frac{1}{2} \times (30 + 14) \times h\)

\( \Rightarrow h = \frac{{440 \times 2}}{{44}}\;{\rm{cm}}\)

\( \Rightarrow h = 10 \times 2\;{\rm{cm}}\)

\( \Rightarrow h = 20\;{\rm{cm}}\)

Therefore, the height of the trapezium is \(20\;{\rm{cm}}\)

** Q.3. The area of the trapezium-shaped field is \(480\;{{\rm{m}}^2}\), the height is \({\rm{15\,m}}\) and one of the parallel sides is \(20\,{\rm{m}}.\) Find the other side.** Let \(ABCD\) be the trapezium and \(AL\) be the height.

Ans:

Given: Area of a trapezium \( = 480\;{{\rm{m}}^2}\)

We know that, area of a trapezium

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

\( \Rightarrow 480 = \frac{1}{2} \times (AB + CD) \times AL\)

\( \Rightarrow 480 = \frac{1}{2} \times (20 + CD) \times 15\)

\( \Rightarrow 20 + CD = \frac{{480 \times 2}}{{15}}\)

\( \Rightarrow 20 + CD = 32 \times 2\)

\( \Rightarrow 20 + CD = 64\)

\( \Rightarrow CD = 44\;{\rm{cm}}\)

Therefore, the other side of the given trapezium is \(44\;{\rm{cm}}\)

** Q.4. Find the sum of the lengths of the bases of a trapezium whose altitude is \(11\;{\rm{cm}}\) and whose area is \(0.55\;{{\rm{m}}^2}\)** Given: Altitude of the trapezium \( = 11\;{\rm{cm}} = \frac{{11}}{{100}}\;{\rm{m}}\) and area of the trapezium \( = 0.55\;{{\rm{m}}^2}\).

Ans:

We know that, area of a trapezium

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

\( \Rightarrow 0.55 = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

\( \Rightarrow \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \frac{{11}}{{100}} = 0.55\)

\( \Rightarrow {\rm{the\;sum\;of\;the\;parallel\;sides}} = \frac{{0.55 \times 200}}{{11}}{\rm{m}} = 10\,{\rm{m}}\)

Therefore, the sum of the length of the parallel sides is \({\rm{10\,m}}\).

** Q.5. The area of the trapezium is \(105\,{\rm{c}}{{\rm{m}}^2}\) and its height is \(7\,{\rm{cm}}.\) If one of the parallel sides is longer than the other by \(6\,{\rm{cm}}\) find the two parallel sides.** Let the length of the smaller side be \(x\,{\rm{cm}}.\) Then, the length of the other side is \(\left( {x + 6} \right)\,{\rm{cm}}.\)

Ans:

Given: Height of the trapezium \( = 7\;{\rm{cm}}\) and area of the trapezium is \(105\,{\rm{c}}{{\rm{m}}^2}\)

We know that, area of a trapezium

\( = \frac{1}{2} \times \left( {{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}} \right) \times \left( {{\rm{Distance}}\,{\rm{between}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}} \right)\)

\( \Rightarrow 105 = \frac{1}{2} \times \left( {{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{parallel}}\,{\rm{sides}}} \right) \times 7\)

\( \Rightarrow 105 = \frac{1}{2} \times (x + 6 + x) \times 7\)

\( \Rightarrow 2x + 6 = \frac{{105 \times 2}}{7}\)

\( \Rightarrow 2x + 6 = 30\)

\( \Rightarrow 2x = 24\)

\( \Rightarrow x = 12\;{\rm{cm}}\)

Therefore, the length of the parallel sides are \(12\,{\rm{cm}}\) and \(\left( {12 + 6} \right)\,{\rm{cm = 18}}\,{\rm{cm}}\)

In the above article, we have studied the definition of trapezium, the properties of the trapezium and the formula to find the trapezium area. Also, we have derived the formula for the area of the trapezium using triangles and using two trapezoids.

**Learn All the Concepts on Trapezium**

** Q.1. What are the properties of trapezium?**The properties of a trapezoid are:

Ans:

1. Like other quadrilaterals, the sum of all the four angles of the trapezium is equal to \({360^ \circ }\).

2. A trapezium has two parallel sides and two non-parallel sides.

3. The diagonals of isosceles trapezium bisect each other.

4. The length of the mid-segment is equal to half the sum of the parallel bases in a trapezium.

5. Two pairs of adjacent angles of a trapezium formed between the parallel sides, and one of the non-parallel sides are supplementary.

** Q.2. What are the formulas for the area and perimeter of a trapezoid?**The area of a trapezoid is given by

Ans:

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\) and the perimeter of a trapezoid \(=a+b+c+d\)

Where, \(a, b, c\) and \(d\) are the length of the sides of the trapezium.

** Q.3. What is the relationship between the diagonals of a trapezium?**The diagonals of isosceles trapezium bisect each other. The length of the mid-segment is equal to half the sum of the parallel bases in a trapezium. Two pairs of adjacent angles of a trapezium formed between the parallel sides, and one of the non-parallel sides, are supplementary.

Ans:

** Q.4. What is an isosceles and right-sided trapezoid?** A right trapezoid (also called a right-angled trapezoid) has two adjacent right angles.

Ans:

A trapezium in which the two non-parallel sides are equal and form equal angles at one of the bases is an isosceles trapezoid.

** Q.5.What is the formula for the area of a trapezoid?** The area of a trapezoid is given by

Ans:

\( = \frac{1}{2} \times \left( {{\rm{sum\;of\;the\;parallel\;sides}}} \right) \times \left( {{\rm{Distance\;between\;the\;parallel\;sides}}} \right)\)

*We hope you find this detailed article on the area of a trapezoid formula helpful. If you have any doubts or queries regarding this topic, feel free to ask us in the comment section and we will be more than happy to assist you. Happy learning!*

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