What is the difference between a square and a rhombus?

A square has equal sides and 90° angles, while a rhombus has equal sides but not necessarily 90° angles.

How do you construct a quadrilateral?

To construct a quadrilateral, you should know the length of the sides and the angle or use coordinates to plot the vertices.

Can a quadrilateral have unequal angles?

Yes, angles of a quadrilateral can be unequal except for special quadrilaterals like squares and rectangles, in which all the angles are equal.

What are the diagonals of a rhombus related to?

Diagonals of a rhombus bisect each other at right angles and divide the rhombus into four right-angled triangles.

Quadrilaterals: Types, Properties & Formulas

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Quadrilateral

The word “Quadrilateral” is taken from the two different words of the Latin language in which “Quadri” means four and lateral means sides, showcasing the fact that it has four sides. The interesting fact about a quadrilateral is that it includes a wide range of different shapes with various properties and not necessarily with all equal sides.

1.0Quadrilateral Definition in Mathematics

The Quadrilateral is a two-dimensional geometric polygon with four sides, vertices, and four angles. The quadrilateral is constructed by joining the set of four points, for instance, A, B, C, and D, in a two-dimensional plane forming the sides AB, BC, CD, and DA and angles ∠A, ∠B, ∠C, and ∠D.

The diagonals of the quadrilateral are formed by joining the opposite vertices; in quadrilateral ABCD, the diagonals are AC and BD.

2.0Properties of Quadrilaterals

The properties of quadrilaterals differ with different shapes of the quadrilaterals, that is, rectangle, square, trapezium, etc. Here are some general properties of a quadrilateral, which remain constant for every quadrilateral shape:

All quadrilaterals have four sides, four vertices, and four angles.
The sum of the interior angles of any quadrilateral is always equal to 360°.
All the quadrilaterals have only two diagonals connecting the opposite side of the quadrilaterals.
Quadrilaterals with all the interior angles less than 180° are known as the convex quadrilaterals. On the other hand, quadrilaterals with at least one interior angle greater than 180° are called concave quadrilaterals.
The sum of all the exterior angles of any quadrilateral is also equal to 360°.

3.0Quadrilateral Types

Quadrilaterals, based on different properties and shapes, are divided into different types of quadrilaterals. Which includes:

Properties of different Quadrilaterals	Quadrilateral Shapes
1. Parallelogram: The opposite sides of any parallelogram are always equal in length. Moreover, the opposite angles of any parallelogram are also equal. The diagonals bisect each other.
2. Rectangle: It is a special type of parallelogram. Hence, possesses all the properties of a parallelogram in addition to others. The measure of all angles is 90°. The diagonals of a rectangle are equal to each other.
3. Square: It is a special type of rectangle as well as a parallelogram. All the sides of the square are equal All the angles of a rectangle are also equal and are 90°. The lengths of the diagonals are also equal to each other.
4. Rhombus: A type of parallelogram with all sides equal to each other. Opposite angles are equal. Diagonals bisect each other at 90° but are not equal to each other.
5. Trapezium: Only one pair of opposite sides is parallel, and the other side remains non-parallel. The diagonals are neither equal nor bisected to each other.
6. Kite: A kite has 2 pairs of adjacent sides that are equal in length. Any one pair of opposite angles are equal. The diagonals intersect at right angles, with a larger diagonal bisecting the smaller one.

4.0Formulas Related to Quadrilaterals

The two types of basic formulas of quadrilaterals include the Area and Perimeter of all the quadrilaterals. The Formula for each which are:

Quadrilateral Name	Perimeter	Area
Rectangle	$2 (l e n g t h + b re a d t h)$	$l e n g t h \times b re a d t h$
Square	$4 \times s i d e$	$S i d e^{2}$
Parallelogram	$2 (B a se + S i d e)$	$B a se \times He i g h t$
Rhombus	$4 \times s i d e$	$\frac{1}{2} d ia g o na l 1 \times d ia g o na l 2$
Trapezium	The sum of all sides	$\frac{1}{2} (s u m o f p a r a ll e l s i d es) h e i g h t$
Kite	$2 (s u m o f a d ja ce n t s i d es)$	$\frac{1}{2} d ia g o na l 1 \times d ia g o na l 2$

5.0Solved Problems

Problem 1: A rectangular garden having a length of 15m and a width of 10m. The gardener wants to lay out grass over the entire garden. How much area does he need to cover with grass?

Solution: According to question length = 15m and breadth = 10m

Area of rectangular garden = $l e n g t h \times B re a d t h$

= $15 \times 10 = 150 m^{2}$

Hence, the Area he needs to cover with grass is 150m²

Problem 2: In a trapezium, the lengths of the parallel sides are 10 cm and 14 cm. The distance between the parallel sides (height) is 8 cm. Find the area of the trapezium.

Solution: According to the question

length of the parallel side (1) let a = 10cm

length of the parallel side (2) let b = 14cm

Height between parallel sides = 8cm

The Area of the Trapezium = $\frac{1}{2} (s u m o f p a r a ll e l s i d es) h e i g h t$

= $\frac{1}{2} (10 + 14) 8 = 24 \times 4 = 96 c m^{2}$

Problem 3: A rhombus has one diagonal that is 12 cm longer than the other. The area of the rhombus is 144 cm². Find the lengths of both diagonals.

Solution: According to the question,

The area of the rhombus = 144 cm²

$\frac{1}{2} d ia g o na l 1 \times d ia g o na l 2 = 144$

$\frac{1}{2} \times 12 \times d ia g o na l (2) = 144$

$6 \times d ia g o na l (2) = 144$

$d ia g o na l 2 = 144/6 = 24 c m$

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