Area of Quadrilateral

Quadrilateral

A quadrilateral is a plave closed figure with four sides and four vertices. It is a four sided polygon.
A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal. In the figure below
quadrilateral \(ABCD\) has two diagonals \(AC\) and \(BD\).

Types of quadrilaterals

There are six basic types of quadrilaterals.

1. Square
   all angles \(90^º\), all sides equal
2. Rectangle
   all angles are 90º
3. Parallelogram
   opposite sides are parallel
4. Rhombus
   all sides are equal
5. Trapezium
   a pair of opposite sides parallel
6. Kite
   two pairs of adjacent sides are equal

Rectangle

A rectangle is a quadrilateral with opposite sides equal and all angles equal to \(90\) degrees.

Area of rectangle

To find the area of a rectangle, we use the formula
\(\square= bh\)
This means we multiply the length of the rectangle by its breadth (width).

NOTE: Drag the points A or C.

This is a rectangle, it has

• opposite sides equal
• each angle equal to \(90^0\)
• diagonals are equal
• diagonals bisect each other

Every rectangle is a parallelogram.

Rectangle Area Challenge

Drag points A or C to change the rectangle. Calculate the area and enter your answer!

Enter the area:

Square

A square is a quadrilateral with all sides are equal and all angles equal to \(90\) degrees.

Area of square

To find its area, we use the formula
\(\square= l^2\)
Here, the length is any one of its sides.

NOTE: Drag the points A or B.

This is a square, it has

• all sides equal
• each angle equal to \(90^0\)
• diagonals are equal
• diagonals bisect each other
• diagonals are perpendicular

Every square is a rectaangle.

Parallelogram

A parallelogram is a quadrilateral in which opposite sides are equal and parallel.

Area of parallelogram

To find its area, we use the formula
\(\square= bh\)
Here, the base is any one of its sides, and the height is the perpendicular distance from the base to the opposite side (not the slanted side).

NOTE: Drag the points A or C

This formula
\(\square= bh\)
works because a parallelogram can be rearranged into a rectangle without changing its area.

NOTE: Drag the point C

Rhombus

A rhombus is a quadrilateral in which all sides are equal and opposite sides are parallel.

Area of rhombus

To find its area, we use the formula
\(\square= bh\)
Here, the base is any one of its sides, and the height is the perpendicular distance from the base to the opposite side (not the slanted side).

NOTE: Drag the points A or C

This formula
\(\square= bh\)
works because a parallelogram can be rearranged into a rectangle without changing its area.

NOTE: Drag the point C

Trapezium

A trapezium (also called a trapezoid) is a quadrilateral with one pair of opposite sides that are parallel, called the bases.

Area of trapezium

To find its area, we use the formula
\(\square= \frac{1}{2} h\) (sum of parallel sides)
\(\square= \frac{1}{2}h (l_1+l_2)\)
where \(l_1\) and \(l_2\) are the lengths of the two parallel sides, and \(h\) is the height, the perpendicular distance between them.

NOTE: Drag the points A or B or C or D

This formula works because a trapezium can be thought of as a combination of two triangles having the base at parallel sides.

Area of trapezium \(=\triangle _1+\triangle _2= \textcolor{blue}{(\frac{1}{2} h \times l_1)}+\textcolor{red}{(\frac{1}{2} h \times l_2)}=\frac{1}{2} h \times (l_1+l_2)\)

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal and one pair of opposite angles equal.

Area of kite

What is the formula for area of kite?

To find the area of a kite, we use the formula
\(\square= \frac{1}{2}(d_1 \cdot d_2)\)
where \(d_1\) and \(d_2\) are the lengths of the two diagonals. The diagonals of a kite intersect at right angles \((90^°)\), and one of them bisects the other.

NOTE: Drag the points A or B or C or D

This special property allows us to divide the kite into two congruent triangles.
By finding the area of these triangles and combining them, we get the area of the kite.

Area of kite \(=\triangle _1+\triangle _2=\) \(\frac{1}{2} \times d_2 \times \frac{1}{2} d_1\)\(+\textcolor{red}{\frac{1}{2} \times d_2 \times \frac{1}{2} d_1}=\frac{1}{2} (d_1+d_2)\)

Area of rhombus, square, kite

समलम्ब चतुर्भुज (rhombus), वर्ग (square), र काईट (kite) तीनवटै चतुर्भुजहरुको क्षेत्रफल एकै प्रकारको सूत्रबाट, विकर्णहरूको प्रयोग गरेर पनि पत्ता लगाउन सकिन्छ।
\(\square= \frac{1}{2}(d_1 \cdot d_2)\)

The properties of quadrilaterals

	Parallelogram	Rectangle	Rhombus	Square	Kite
Both pairs of opposite sides parallel	X	X	X	X
Both pairs of opposite sides congruent	X	X	X	X
All sides congruent			X	X
Opposite angles congruent	X	X	X	X
Diagonals perpendicular			X	X	X

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