What is a Quadrilateral?
A quadrilateral is a polygon with 4 sides and 4 corners or vertices. Depending on the length of the sides and the measure of the angles, quadrilaterals are classified as rectangles, squares, parallelograms, trapezoids and kites.
What is the Area of a Quadrilateral?
Let’s learn about the area of different types of quadrilaterals.
- Area of a Rectangle
Rectangles are quadrilaterals in which the opposite sides are congruent and parallel to each other. Each of the interior angles is a right angle. We can find the area of a rectangle by multiplying the length with the width of any given rectangle.
Area of Rectangle, \(A =lw\)
- Area of a Square
A square is a type of quadrilateral with all sides equal in length, and each angle equal to 90°. Another feature of a square is that the opposite sides of every square are parallel. The area of a square is side length times side length.
Area of Square, \(A = s^{2}\)
- Area of a Parallelogram
Parallelogram is a quadrilateral with equal and opposite parallel sides. When we multiply the base and height of a parallelogram, we obtain its area.
Area of Parallelogram, \(A =\text{ }bh\)
- Area of a Trapezoid
A trapezoid is a quadrilateral having one pair of opposite sides that are parallel to each other. In the case of a trapezoid the area is calculated by applying the formula, half the product of height and the sum of the length of the parallel sides, which are called bases.
Area of a Trapezoid, \(A = \frac{1}{2}h\text{ }\left( b_{1} + b_{2}\right)\)
- Area of Kite
Kites are quadrilaterals with two pairs of adjacent sides equal in length and two pairs of adjacent sides unequal in length. When referring to a shape like the kite, the area is found by applying the formula which is stated as, sum of the area of the two triangles formed by the diagonals of the kite.
Area of a kite, \(A = \frac{1}{2}h \left( b+a \right)+\frac{1}{2}H\left( b+a \right)\)
\(=\frac{1}{2}\left( h+H \right)\left( b+a \right)\)
Here, (h+H) and (b+a) are the measures of the two diagonals of the kite, so,
Area of a kite, \(A = \frac{1}{2}\text{ }d_{1}d_{2}\)
Solved Examples
Example 1: What is the area of the trapezoid with the measurements provided in the image?
Solution:
\(A =\frac{1}{2}h\left( b_{1} + b_{2}\right)\) [Write the formula]
\( = \frac{1}{2}\left( 18 \right)\left( 33+44 \right)\) [Substitute 18 for h, 33 of \(b_{1}\) and 44 for \(b_{2}\)]
\(=\frac{1}{2}\left( 18 \right)\left( 77 \right)\) [Add 44 and 33]
\(=693\text{ }sq.\text{ }cm.\) [Simplify]
So, the area of the trapezoid is 683 square centimeters.
Example 2: What is the height of a parallelogram with an area of 144 square feet and with a base length of 24 feet?
Solution:
\(A = bh\)
\(144 = 24 \times h\) [Substitute 144 for A and 24 for b]
\(\frac{144}{24}=h\) [Divide both sides by 24]
\(6=h\) [Simplify]
\(h=6\text{ }ft\) [Rewrite equation]
So, the height of the parallelogram is 6 feet.
Example 3: John, William and Alex have three pieces of land in the shape of different quadrilaterals. The piece of land that John has is in the shape of a rectangle, William’s piece of land is shaped like a trapezoid, and the land that belongs to Alex is shaped like a square. Find the total area of all the pieces of land. Measurements are provided in the image.
Solution:
Stated in the image:
John, rectangular piece of land,
Length = 25 m
Width = 40 m
Alex, trapezoidal piece of land,
Parallel sides = 40 m and 15 m
Height = 30 m
William, square shaped land,
Side length = 25 m
Total area = Sum of the areas covered by each piece of land
So, Total area = Area of square shaped land + Area of rectangular land + Area of trapezoid-shaped land
\(A = s^{2}+lw+\frac{1}{2}h\left( b_{1} + b_{2} \right)\) [Write the formula for the area]
\(= 25^{2}+25 \times 40 + \frac{1}{2}\times 30\left( 40+15 \right)\) [Substitute the values]
\(= 625 + 1000 + 825\) [Simplify]
\(= 2450\text{ }sq.m\) [Add]
So, the total area of the three pieces of land is 2450 square meters.
The non-parallel opposite sides of a trapezoid are known as legs.
The angle sum property of a quadrilateral states that the sum of all the interior angles of the quadrilateral is always 360 degrees.
A polygon is a closed geometric shape formed by line segments.
The perimeter of a shape is the total length of its boundary, and its area is the space enclosed within the boundary.