Perimeter of Quadrilateral

The perimeter of quadrilateral is the total length of its boundary. A quadrilateral is a four-sided polygon that can be regular or irregular. In a regular quadrilateral, all the sides are equal in length and all the angles are of equal measure, whereas, in an irregular quadrilateral, the sides and angles are not equal. There are 6 specific types of quadrilaterals - Square, rectangle, parallelogram, rhombus, kite, and trapezoid. Let us learn how to find the perimeter of quadrilateral in this page.

The perimeter of a quadrilateral is the length of its boundary, i.e., if we join all the four sides of a quadrilateral to form a single line segment, the length of the resultant line segment is called its perimeter. Thus, the unit of the perimeter of a quadrilateral is the same as that of its side, i.e., it is measured in linear units like meters, inches, centimeters, etc.

We know that the perimeter of a quadrilateral can be obtained by adding all its side lengths. This can be expressed by a simple formula. For example, the formula for the perimeter of a quadrilateral ABCD can be expressed as,

Perimeter = AB + BC + CD + DA

We have already seen that there are 6 specific types of quadrilaterals, which are, square, rectangle, parallelogram, rhombus, kite, and trapezoid. Though the perimeter of a quadrilateral is the sum of all its sides, sometimes, all the side lengths might not have been given. In such cases, we need to recollect the properties of quadrilaterals with respect to sides, in order to obtain the side lengths that are not given. For example, if we need to find the perimeter of a square with only one side length given, we need to recollect one of the properties of the square, that all its side lengths are equal. So, if we assume one side of the square to be x, its perimeter will be x + x + x + x = 4x. In the same way, we can derive the perimeter formulas of all of the 6 specific types of quadrilaterals. Observe the following figure to see the different formulas that are used for calculating the perimeter of quadrilaterals.

Sometimes, a quadrilateral has a circle inside it. This is termed as a circumscribed quadrilateral or a quadrilateral with an inscribed circle. In such cases, we use the property of the tangent of a circle which says "any two tangents drawn to a circle from a point are of equal lengths". We will see how to find the perimeter of a circumscribed quadrilateral (or) the perimeter of a quadrilateral with a circle inside it using the example given below.

Example: Find the perimeter of the following quadrilateral.

Solution:

Using the property of tangents - 'Any two tangents drawn to a circle from a point are of equal lengths', let us find the perimeter of quadrilateral with an inscribed circle.

PT = PU = 5 inches

QV = QU = 2 inches

RW = RV = 3 inches

ST = SW = 4 inches

Now the perimeter of the quadrilateral is,

PQ + QR + RS + SP

= (PU + UQ) + (QV + VR) + (RW + WS) + (ST + TP)

= (5 + 2) + (2 + 3) + (3 + 4) + (4 + 5)

= 28 inches

Therefore, the perimeter of the given quadrilateral = 28 inches.

Note: We can use the same property of tangents of a circle "two tangents drawn to a circle from a point are of equal lengths" to find the perimeter of a cyclic quadrilateral (a quadrilateral that is inscribed in a circle) as well.

What is Meant by the Perimeter of Quadrilateral?

The perimeter of a quadrilateral is the total length of its boundary. For example, the perimeter of a quadrilateral ABCD can be expressed as, Perimeter = AB + BC + CD + DA. This means if all the sides of a quadrilateral are known, we can get its perimeter by adding all its sides.

What is the Formula of Perimeter of Quadrilateral?

The basic formula that is used to find the perimeter of a quadrilateral is, Perimeter = a + b + c + d, where a, b, c, and d are the four sides of the quadrilateral. Although this formula is used for all quadrilaterals, for some quadrilaterals like square and rectangle, this formula can be simplified because in a square, all the sides are equal, so the perimeter formula becomes, a + a + a + a = 4a. Similarly, in a rectangle, the opposite sides are equal, so the formula changes to 2(length + width). However, the basic formula for the perimeter of all quadrilaterals remains the same.

How to Find the Perimeter of a Quadrilateral With a Missing Side?

The missing side of a quadrilateral can be found by using the properties of quadrilaterals. Once all the missing sides are found, we can find the perimeter of the quadrilateral by adding all the side lengths. For example, if we know one side of a square, we can find the other 3 sides because we know that all the sides of a square are equal. After that, we can find the perimeter.

How to Find the Perimeter of Quadrilateral With Inscribed Circle?

When a quadrilateral has an inscribed circle, the sides of the quadrilateral become the tangents to the circle. Therefore, the property of tangents can be used to find the perimeter of the given quadrilateral. The property of tangent states that - 'Any two tangents drawn from a point to some points on a circle are always of equal lengths'. This property helps to find the missing side lengths of the quadrilateral. Once all the missing sides are found, we can find the perimeter of the quadrilateral by adding all four sides.

How to Find the Perimeter of a Quadrilateral With Coordinates?

When the coordinates of a quadrilateral's vertices are given, we first find its side lengths by using the distance formula. Then, we add all the four sides to find its perimeter.

How to Find the Perimeter of a Quadrilateral on Graph?

To find the perimeter of a quadrilateral on a graph, we use the following steps.

• Step 1: Find the coordinates of the vertices from the graph.
• Step 2: Apply the distance formula to find the side lengths of the quadrilateral.
• Step 3: Add all the side lengths and the sum gives the perimeter of the quadrilateral.

How to Find the Perimeter of Quadrilateral that is Inscribed in a Circle?

When a quadrilateral is inscribed in a circle, it is called a cyclic quadrilateral. To find the missing sides of a cyclic quadrilateral, we apply the property of the tangent of a circle that says, 'Any two tangents of a circle drawn from the same point are of the same lengths'. After finding the missing sides, we add all the four sides to find the perimeter.