Quadrilaterals

< 23.8 Pythagoras Theorem – A Visual Proof | Topic Index | 23.10 Types of Quadrilaterals >

Quadrilaterals which are closed figures with 4 sides, are the next simplest shapes after triangles. Unlike triangles, quadrilaterals are not rigid structures.

Angle Sum

Any quadrilateral can be divided into 2 triangles by drawing any of the two diagonals. Hence the total of the interior angles of a quadrilateral is the sum of the interior angles of the 2 triangles into which it has been divided, which is 360 degrees.

Types of Quadrilaterals

Convex &amp; Concave Quadrilaterals

All quadrilaterals can be classified into concave &amp; convex. In a concave quadrilateral, one of the internal angles will be a reflex angle (&gt;180). They roughly resemble the tip of an arrow. They are not studied in detail in school geometry.

In a convex quadrilateral all the internal angles are less than a straight angle.

Types of convex quadrilaterals

There are several well-known types of convex quadrilaterals. A major classification of quadrilaterals is Trapezium &amp; Kite.

Properties &amp; relations of various types of quadrilaterals. Chapter 23.10 give a visual representation which would make understanding easier and make memorization unnecessary.

Trapezium - If any 2 opposing sides of the quadrilateral are parallel, it is a Trapezium. Trapezium was studied well in ancient cultures. It resembles the cross section of a canal. All cultures depended upon water to transport heavy loads and had to dig canals. The volume of earth to be dug out depended on its cross-sectional area. Hence the formula for the area of a trapezium was worked out in ancient cultures.

Kite - If on the other hand, in a quadrilateral, both pairs of adjacent sides are equal, it is a Kite. The diagonals of a Kite are perpendicular to each other. A Kite can be divided by one of the diagonals into 2 isosceles triangles, which have the same 3rd(non-equal) side. The other diagonal divides a Kite into 2 congruent triangles.

If the other 2 sides of the Trapezium are also parallel then we get a Parallelogram. We will study Parallelograms and shapes related to them in the next chapter.

Rhombus - In the next chapter, we will also come across the Rhombus, which belongs both the Parallelogram &amp; Kite families. If in a Kite, the opposing sides are equal, then also we get a Rhombus.

Chapter 23.10 is arranged like a family tree, giving the relations between various kinds of quadrilaterals in a logical manner.

We can say a Rhombus is a Kite which is a Quadrilateral. We can also say a Rhombus is a Parallelogram which is a Trapezium which is a Quadrilateral.

Area - The area of a quadrilateral can be measured by dividing it into 2 triangles whose area can be worked out individually. In various types of quadrilaterals, because of their symmetry, simple formulas can be worked out for finding the area.

Cyclic Quadrilaterals 

If a circle can be drawn through the four vertices of a quadrilateral, it is called a Cyclic Quadrilateral. Or we can say that, if any four points on the circumference of a circle are joined to form a quadrilateral, it would be a cyclic quadrilateral. A cyclic quadrilateral has some special properties.

The opposite angles of a cyclic quadrilateral will be supplementary or total to 180 degrees.

Conversely, we can say that if the opposite angles of a quadrilateral add up to 180 degrees, then a circle can be drawn passing through all the four vertices. A square and a rectangle are examples of such quadrilaterals.

The proof of this can be understood after we study properties of a circle.

< 23.8 Pythagoras Theorem – A Visual Proof | Topic Index | 23.10 Types of Quadrilaterals >