Welcome to this article on the area of quadrilaterals. A quadrilateral is a closed two-dimensional geometric shape that has four sides and four vertices. Examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and rhombuses. The area of a quadrilateral is the measure of the region enclosed by its four sides, and it is expressed in square units.

What is a Quadrilateral?

A quadrilateral is a four-sided polygon. It is a closed shape with four straight sides and four vertices. Some common examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and kites.

Types of Quadrilaterals

Properties of Quadrilateral:

The properties of a quadrilateral vary depending on the specific type of quadrilateral. However, there are some general properties that apply to all quadrilaterals:

  • Four sides: Quadrilaterals have four sides, which are connected by four vertices.
  • Interior angles: The sum of the interior angles of a quadrilateral is always equal to 360 degrees.
  • Diagonals: A diagonal is a line segment that connects two non-adjacent vertices of a quadrilateral. The number of diagonals in a quadrilateral depends on the number of vertices: a quadrilateral with n vertices has n(n-3)/2 diagonals.
  • Opposite sides are parallel: In a parallelogram, the opposite sides are parallel to each other.
  • Opposite angles are congruent: In a parallelogram, the opposite angles are congruent (i.e., have the same measure).
  • Consecutive angles are supplementary: In a parallelogram, the consecutive angles are supplementary (i.e., add up to 180 degrees).
  • Diagonals bisect each other: In a parallelogram, the diagonals bisect each other (i.e., they divide each other into two equal parts).
  • Equal opposite sides and angles: In a rhombus, all four sides are equal in length and all four angles are congruent (i.e., have the same measure).
  • Perpendicular diagonals: In a square, the diagonals are perpendicular to each other.
  • Right angles: In a rectangle, all four angles are right angles (i.e., have a measure of 90 degrees).
  • Parallel sides and unequal adjacent sides: In a trapezoid, the two opposite sides are parallel to each other, and the two adjacent sides are unequal in length.

Area of Quadrilateral Formula:

To calculate the area of a quadrilateral, we need to know its base and height. However, not all quadrilaterals have a straight base and height. In this case, we can use a formula that involves the lengths of the four sides and the angle between them. This formula is called Brahmagupta’s formula, named after the Indian mathematician Brahmagupta who discovered it in the 7th century AD.

The formula for the area of a quadrilateral using Brahmagupta’s formula is: Area = √(s-a)(s-b)(s-c)(s-d) where s = (a+b+c+d)/2 is the semiperimeter of the quadrilateral, and a, b, c, and d are the lengths of its four sides.

Area of Special Quadrilaterals:

Different types of quadrilaterals have specific formulas for their areas. Let us look at some examples:

  • Square: The area of a square is given by the formula A = a^2, where a is the length of its side.
  • Rectangle: The area of a rectangle is given by the formula A = lw, where l is the length and w is the width of the rectangle.
  • Parallelogram: The area of a parallelogram is given by the formula A = bh, where b is the base and h is the height of the parallelogram.
  • Trapezoid: The area of a trapezoid is given by the formula A = (b1+b2)/2 * h, where b1 and b2 are the lengths of the parallel bases and h is the height of the trapezoid.
  • Rhombus: The area of a rhombus is given by the formula A = (d1*d2)/2, where d1 and d2 are the lengths of the diagonals of the rhombus.
Sr. No. Quadrilateral Formula for Area
1. Square A = s^2
2. Rectangle A = lw
3. Parallelogram A = bh
4. Trapezium A = (a + b)h/2
5. Kite A = (d1 x d2) / 2
6. Rhombus A = (d1 x d2) / 2

where:

  • s is the length of a side of the square
  • l is the length of the rectangle
  • w is the width of the rectangle
  • b is the base of the parallelogram
  • h is the height of the parallelogram or trapezium
  • d1 and d2 are the diagonals of the kite or rhombus
  • a and b are the lengths of the parallel sides of the trapezium, and h is the height.

Applications of Quadrilaterals:

Quadrilaterals have practical applications in many areas of life. Architects and engineers use quadrilaterals to design buildings and structures. Surveyors use quadrilaterals to determine the area of a piece of land. Artists use quadrilaterals to create art and designs. Quadrilaterals are also used in the construction of roads, bridges, and other infrastructure.

FAQs:

Can a quadrilateral have an area of 0?

Yes, a quadrilateral can have an area of 0. This occurs when its sides are collinear, meaning they lie on the same line.

How do I find the height of a trapezoid?

To find the height of a trapezoid, you can use the formula h = (2A)/(b1+b2), where A is the area of the trapezoid and b1 and b2 are the lengths of its parallel bases.

What is the difference between a quadrilateral and a polygon?

A quadrilateral is a type of polygon that has four sides and four vertices. A polygon is a closed two-dimensional shape that has three or more sides and vertices.

In conclusion, the area of a quadrilateral is an important concept in geometry that is used in various fields. It is essential to understand the formulas for finding the areas of different types of quadrilaterals to apply them in real-life situations.

Solve Examples:

Example 1: Finding the area of a square

Given the length of a side of a square is 8 cm, find its area.

Solution:

The formula for the area of a square is A = s^2, where s is the length of a side.

Using this formula, we get:

A = 8^2 = 64 square cm

Therefore, the area of the square is 64 square cm.

Example 2: Finding the area of a parallelogram

Given the base of a parallelogram is 12 cm and the height is 8 cm, find its area.

Solution:

The formula for the area of a parallelogram is A = bh, where b is the base and h is the height.

Using this formula, we get:

A = 12 x 8 = 96 square cm

Therefore, the area of the parallelogram is 96 square cm.

Example 3: Finding the area of a trapezium

Given the lengths of the parallel sides of a trapezium are 5 cm and 10 cm, and the height is 6 cm, find its area.

Solution:

The formula for the area of a trapezium is A = (a + b)h/2, where a and b are the lengths of the parallel sides and h is the height.

Using this formula, we get:

A = (5 + 10) x 6/2 = 45 square cm

Therefore, the area of the trapezium is 45 square cm.

Example 4: Finding the area of a kite

1) Given the lengths of the diagonals of a kite are 6 cm and 8 cm, find its area.

Solution:

The formula for the area of a kite is A = (d1 x d2)/2, where d1 and d2 are the lengths of the diagonals.

Using this formula, we get:

A = (6 x 8)/2 = 24 square cm

Therefore, the area of the kite is 24 square cm.

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