Last modified on April 22nd, 2021

chapter outline

 

Regular and Irregular Quadrilaterals

Based on their sides, all quadrilaterals can be broadly classified into two different groups: regular and irregular.

Regular and Irregular Quadrilateral

What is a Regular Quadrilateral

A regular quadrilateral is a type of quadrilateral with four sides of equal length and four angles of equal measure.

Example: A square is the only regular quadrilateral

Regular Quadrilateral

What is an Irregular Quadrilateral

An irregular quadrilateral is a type of quadrilateral having one or more sides of unequal length and one or more angles of unequal measure.

Examples: Rectangle, parallelogram, rhombus, trapezoid, and kite are irregular quadrilaterals.

Irregular Quadrilateral

Formulas

Area of Regular Quadrilateral

A square is the only regular quadrilateral known to us. Thus, to find the area of a regular quadrilateral we will use the formula for determining the area of a square. The formula is given below:

Formula

Area (A) = a2, here a = side

In regular quadrilateral ABCD,

a = AB = BC = CD = DA

Find the area of the regular quadrilateral ABCD whose sides measure 12 m.

Solution:

As we know,
Area (A) = a2, here a = 12 m
= 12 x 12 m2
= 144 m2

Area of Irregular Quadrilateral

Since all irregular quadrilaterals are different in shape, we cannot apply the formula of any particular quadrilateral to find the area of all irregular quadrilaterals. In other words, we do not have a fixed, general formula that can be used for all of them. In such cases the following steps are followed:

  • The quadrilateral is divided into two triangles by drawing a diagonal
  • Find the area of each triangle using different formulas as described in our triangles article
  • Add the areas of two triangles

The steps though sounds easy can sometimes be quite elaborate. Let us take an example to understand the concepts better.

Find the area of an irregular quadrilateral ABCD when sides BC = 6cm, CD = 8 cm, DA = 10 cm, and AB = 12 cm and ∠BCD = 120°.

Solution:

Let us divide the irregular quadrilateral ABCD into △BCD and △DAB by drawing a diagonal BD
Since we do not know the height of any of the triangles BCD and DAB, we cannot use the general formula ½ x base x height, for determining the area of triangles.
Using Side-Angle-Side (SAS) Law
Here, we will use the trigonometric function (A) = ½ x BC x CD x sin C, to calculate the area of △BCD
A = ½ x 6 x 8 x sin 120°
= ½ x 48 x √3/2 cm2
= 20.78 cm2
Using the Law of Cosines
We now know the area of △ BCD, but still do not know the length of the diagonal BD. For that, we will use the Law of Cosines:
c2 = a2 + b2 – 2ab cos C
In △BCD, let BC = h, CD = a, and BD = t
Then the above law of Cosines can be written as,
t2 = a2 + h2 – 2ah cos T
t2 = (8 x 8) + (6 x 6) – 2 x 8 x 6 cos 120° cm
t2 = 64 + 36 – (96 x cos 120°) cm
t2 = 100 – (96 x (-0.5)) cm
t2 = (100 + 48) cm
t2 = 148 cm
t = 12.165 cm
We now have the approximate length of the diagonal BD as 12.165 cm of the irregular quadrilateral ABCD.
Using Heron’s Formula
Knowing the diagonal of the quadrilateral ABCD, we can now calculate the area of the other section of the quadrilateral using the Heron’s formula.
Since Heron’s formula depends on knowing the semi-perimeter, for △DAB, the three sides are:
DA = 10 cm, AB = 12 cm, and BD = 12.165 cm
As we know, semi-perimeter (s) = ½ (10 + 12 + 12.165)
= 34.165/2 cm
= 17.08 cm
Now, applying Heron’s Formula to calculate the area of △DAB,
A = √s(s – a) (s – b) (s – c), here s = 17.08 cm, a = 10 cm, b = 12.165 cm, and c = 12 cm
= √17.08 x (17.08 – 10) x (17.08 – 12.165) x (17.08 – 12) cm2
= √17.08 x 7.08 x 4. 915 x 5.08 cm2
= √3019.314 cm2
= 54.948 cm2
Now, to obtain the area of the irregular quadrilateral ABCD, we need to add the area of the triangles BCD and DAB
Hence, Area of the irregular quadrilateral ABCD = Area of △BCD + Area of △DAB
= (20.78 + 54.948) cm2
= 75.728 cm2

FAQs

Q1. Name a quadrilateral with four congruent sides that is not regular?

Ans. There is no such quadrilateral that which has four congruent sides but is irregular because the only quadrilateral that has four congruent sides is a square, which is a regular quadrilateral.

Q2. Name an equilateral quadrilateral that is not regular?

Ans. Rhombus

Q3. Is a trapezoid a regular quadrilateral?

Ans. All four sides of a trapezoid are not equal and hence it is not a regular quadrilateral.

Q4. What is the other name of a regular quadrilateral?

Ans. We usually call a regular quadrilateral, a square.

Q5. Which quadrilateral is a regular polygon?

Ans. Square

Q6. A regular quadrilateral has how many lines of symmetry?

Ans. The square being the only regular quadrilateral, it has four lines of symmetry.

Q7. How many sides does a regular quadrilateral have?

Ans. A regular quadrilateral has four sides.

Q8. Is a rectangle a regular quadrilateral?

Ans. Since a rectangle has only opposite sides equal, they are not a regular quadrilateral.

Q9. Is a rhombus a regular quadrilateral?

Ans. Since a rhombus does not have all its four angles equal, they are not a regular quadrilateral.

Q10. Name an equiangular quadrilateral that is not regular?

Ans. Rectangle.

Q11. Can a concave quadrilateral be regular?

Ans. Since, a concave quadrilateral has one of the interior angles measuring more than 90°, it is not possible to have a quadrilateral that is concave but regular.

Q12. Do all irregular quadrilaterals have no right angles?

Ans. The only irregular quadrilateral that has right angles is a rectangle.

Last modified on April 22nd, 2021

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