Quadrilateral is a plane closed figure formed by joining four line segments. It is a polygon with four sides. Quadrilateral has four sides, four vertices and four angles.

Quadrilateral

In the above figure, ABCD is quadrilateral with four line segments AB, BC, CD and DA. AB, BC, CD and DA are called **sides** of Quadrilateral ABCD. Also, A, B, C and D are called **vertices** of quadrilateral.

Quadrilateral ABCD has four angles viz. \(\angle DAB\), \(\angle ABC\), \(\angle BCD\) and \(\angle CDA\). We can write these angles as \(\angle A\), \(\angle B\), \(\angle C\) and \(\angle D\) respectively.

Two sides of a quadrilateral are said to be **adjacent sides** if they meet at a common end point. In above figure, AB and BC are adjacent sides of the quadrilateral ABCD because they have a common end point B.
Also, BC and CD are adjacent sides of the quadrilateral ABCD because they have common end point C. CD and DA are adjacent sides of the quadrilateral ABCD because they have common end point D.
DA and AB are also the adjacent sides of the quadrilateral ABCD because they have common end point A.

The sides of a quadrilateral are called as **opposite sides** if they do not have a common end point. Again, referring to the above quadrilateral ABCD, AB and CD are the opposite sides because they have no common end point.
Also, AD and BC are also the opposite sides of the quadrilateral ABCD because they also have no common end point.

The line segment which joins the opposite vertices of a quadrilateral is called its **diagonal**.

Diagonals of quadrilateral

In the above quadrilateral ABCD, AC and BD are two diagonals of quadrilateral.

**Why?**

Because AC line segment joins two opposite vertices A and C and hence AC is a diagonal of quadrilateral ABCD. Similarly, BD line segment joins two opposite vertices B and D, that makes BD the diagonal of quadrilateral ABCD.

As the number of sides of a quadrilateral are always fixed i.e. four, so, the types of Quadrilateral are classified based on how the length of four sides vary and how inclined the sides are.

The perimeter of any plane figure is total length of its boundary. In other words, we can say perimeter is how long the boundary of a quadrilateral is and is calculated by adding up length of all sides of a quadrilateral. The perimeter is measured by same units as of length i.e. millimeter, centimeter, meter, kilometer etc.

Example

Example of perimeter calculation is how long a runner has to run around a rectangular shape playground to finish one round. The runner has to go around the boundary of the rectangle and cover up all the sides. So, the distance coverd by him will be equal to sum of all sides of rectangle, which is called as perimeter.

The area of any plane figure is the amount of surface enclosed by its sides. Let’s understand it by an example:

Example

Example of area calculation is how much area one has to paint on one side surface of a rectangular shape wooden board. If we need to paint both sides of rectangular board, then it will be two surfaces to paint that will make two areas of board to be painted.

Let’s have a look at very common types of a quadrilateral and how to calculate their perimeter and area.

A quadrilateral, in which both pairs of opposite sides are parallel, is called **parallelogram**. Also, opposite sides of a parallelogram are equal.

Parallelogram

In above figure, ABCD is a parallelogram because opposite sides AB and CD are parallel to each other and have equal length. Also, AD and BC are parallel to each other and have equal length. Or, we can write them as:

AB = CD and AB || CD. Also, AD = BC and AD || BC

Hence, the above ABCD quadrilateral is a **parallelogram**.

**A diagonal of parallelogram divides it into two congruent triangles and the two diagonals of a parallelogram bisect each other.**

Parallelogram

In above parallelogram ABCD, the two diagonals AC and BD bisect each other at O. Therefore, OA=OC and OB=OD.

Parallelogram with two parallel sides a, b and height h

Perimeter of a parallelogram is calculated by adding up length of all sides.

The parallelogram in above figure has two parallel sides a, b and height h. We can write it as:

Perimeter \(= a + b + a + b\) \(= 2a + 2b\) \(= 2(a + b)\)

Formula

Perimeter of parallelogram \(= 2(a + b)\)

Area of a parallelogram is calculated by multiplying its length of base and height h.

So, area of parallelogram \(= base \times height\) \(= b \times h\)

Formula

Area of parallelogram \(= b \times h\)

A quadrilateral in which both pairs of opposite sides are parallel, equal in length and each of four angles is of \(90^0\), is called **rectangle**.

Rectangle

In given figure, ABCD is a rectangle. Because, the opposite sides are parallel, i.e. AB||CD and BC||AD. Secondly, the length of opposite sides are equal, i.e. AB = CD and BC = AD. And the last, all of four angles are of \(90^0\), i.e.

\(\angle A\) = \(90^0\), \(\angle B\) = \(90^0\), \(\angle C\) = \(90^0\), \(\angle D\) = \(90^0\)

**Diagonals of rectangle are always equal in length and bisect each other.**

Rectangle

In above figure, AC = BD, because diagonals of rectangle are equal in length. And OA = OC and OB = OD, because diagonals of rectangle bisect each other.

Rectangle with length l and breadth b

Perimeter of a rectangle is calculated by adding up the length of all sides. The rectangle in above figure has length l and breadth b. Therefore, the perimeter will be the sum of length of its four sides.

We can write it as:

Perimeter \(= l + b + l + b\) \(= 2l + 2b\) \(= 2(l + b)\)

Formula

Perimeter of rectangle \(= 2(l + b)\)

Area of a rectangle is calculated by multiplying its length and breadth. So, area of square \(= length \times breadth\)

\(= l \times b\)

Formula

Area of square \(= l \times b\)

A quadrilateral in which all sides are parallel, equal and each of its angle is 90 degree, is called **square**.

Square

In above figure, ABCD is a square because AB || CD, BC || AD and AB = BC = CD = DA. \(\angle A\) = \(90^0\), \(\angle B\) = \(90^0\), \(\angle C\) = \(90^0\) and \(\angle D\) = \(90^0\).

The diagonals of a square are equal in length and bisect each other at right angle.

Diagonals of square

In above figure, AC and BD are diagonals of the square ABCD and intersect each other at O. Both diagonals are equal in length. i.e. AC = BD.

Both diagonals bisect each other. i.e. OA = OC and OB = OD

Both diagonals intersect each other at right angle. i.e. \(\angle AOB\) = \(\angle BOC\) = \(\angle COD\) = \(\angle AOD\) = \(90^0\)

Square with length l of its four sides

Perimeter of a square is calculated by adding up length of all sides.

The square in above figure has length l of its four sides. Therefore, the perimeter will be the sum of length of its four sides. We can write it as:

Perimeter \(= l + l + l + l = 4l\)

Perimeter of square can also be calculated by multiplying 4 with length l of square. So, perimeter can also be calculated as:

Perimeter \(= 4 \times l = 4l\)

Formula

Perimeter of square \(= 4l\)

Area of a square is calculated by multiplying its length and breadth.

So, area of square \(= length \times breadth\) \(= l \times l\) \(= l^{2}\)

Formula

Area of square \(= l^{2}\)

A quadrilateral in which both pairs of opposite sides are parallel and all four sides are equal in length, is called **rhombus**. Rhombus is also called as **equilateral quadrilateral**.

Rhombus

In above figure, ABCD is a rhombus because AB || CD, BC || AD and AB = BC = CD = DA. **The diagonals of a rhombus bisect each other at right angle.**

Diagonals of rhombus

In above figure, the diagonals AC and BD bisect each other at right angle. i.e. OA = OC and OB = OD. \(\angle AOB\) = \(\angle BOC\) = \(\angle COD\) = \(\angle AOD\) = \(90^0\)

Diagonals of rhombus bisect angles

Also, **in above figure, diagonals of a rhombus bisect the angles of rhombus at vertices.** i.e. \(\angle 1\) = \(\angle 2\), \(\angle 3\) = \(\angle 4\), \(\angle 5\) = \(\angle 6\) and \(\angle 7\) = \(\angle 8\)

Rectangle with length l and breadth b

Perimeter of a rhombus is calculated by adding up length of all sides. The rhombus in above figure has length a of its all sides because length of all sides of a rhombus are equal always. Therefore, the perimeter will be the sum of length of its four sides. We can write it as:

Perimeter \(= a + a + a + a\) \(= 4a\)

Perimeter of rhombus can also be calculated by multiplying 4 with length a. So, perimeter can also be calculated as:

Perimeter \(= 4 \times a\) \(= 4a\)

Rhombus with diagonals

Perimeter of rhombus is also calculated using length of two diagonals.

Perimeter \(= 2\sqrt{d_1^{2} + d_2^{2}}\)

Formula

Perimeter of rhombus \(= 4a\)

Perimeter of rhombus \(= 2\sqrt{d_1^{2} + d_2^{2}}\)

Area of a rhombus is calculated by multiplying its length of base and height h.

So, area of rhombus \(= base \times height\) \(= a x h)

Area of rhombus is also calculated using length of two diagonals.

Area of rhombus \(= \frac{1}{2} \times d1 \times d2\)

Formula

Area of rhombus \(= a \times h\)

Area of rhombus \(= \frac{1}{2} \times d1 \times d2\)

A quadrilateral in which only one pair of opposite sides are parallel, is called **trapezium**.

Trapezium

In above figure, ABCD is a trapezium because AB || CD and AD and BC are non parallel sides.

If two non parallel sides of trapezium are equal, it is called **isosceles trapezium**.

Isosceles Trapezium

In above isosceles trapezium ABCD, AB || CD and non parallel sides BC and AD are equal in length, i.e. BC = AD.

In isosceles trapezium, **base angles are of equal in magnitude**. i.e. \(\angle A\) = \(\angle B\) and \(\angle C\) = \(\angle D\)

In isosceles trapezium, **diagonals are equal in length**. i.e. AC = BD.

Trapezium with sides a, b, c, d and height h

Perimeter of a trapezium is calculated by adding up length of all sides.

The trapezium in above figure has sides a, b, c, d and height h. We can write it as:

Perimeter \(= a + b + c + d\)

Formula

Perimeter of trapezium \(= a + b + c + d\)

Area of trapezium \(= \frac{1}{2}(sum \; of \; parallel \; sides) \times h\) \(= \frac{1}{2}(a + b) \times height\)

Formula

Area of trapezium \(= \frac{1}{2}(a + b) \times h\)

A quadrilateral in which two pairs of adjacent sides are equal, is called **kite**.

Kite

In above figure, ABCD is a kite because AB = BC and CD = DA.