Revision Notes on Area of a Trapezium and Polygon
Mensuration
The part of mathematics in which we measure area, perimeter, and volume of the plane and solid figures is called mensuration.
Area
The surface covered by the border line of the figure is the area of the plain shape.
Unit of the area is square if the length unit.
Perimeter
The length of the boundary of the plane shape is called perimeter.
The unit of the perimeter is same as the length unit.
Area of Trapezium
A quadrilateral whose two sides are parallel is called trapezium.
Note- If its non-parallel sides are equal then it is said to be an isosceles trapezium.
Area of Trapezium can be found,
1. By Splitting the figure
One way to find the Area of trapezium is to divide it into two or three plane figures and then find the area.
In the trapezium ABCD,
It can be divided into two parts i.e. a rectangle and a triangle.
Area of ABCD = Area of ABED + Area of DEC
2. By using formula
Another way is to calculate the area by using formula.
OR, Area of Trapezium=12Sum of parallel sides×height
Area of trapezium is half of the product of the summation of the parallel sides and the perpendicular distance between them.
Example
Find the area of the trapezium whose parallel sides are 6 cm and 16 cm, with a height of 5 cm. Calculate the area using both the methods.
Solution:
Splitting the trapezium we get –
Area of the trapezium = Area of rectangle + Area of a triangle
= (6 x 5) + (1/2) x 5 x 10
= 30 + 25
= 55 cm2
Remark: We should use the formula most of the time if possible as it is the quick and easy method.
Area of a General Quadrilateral
To find the area of any quadrilateral we can divide it into two triangles and then the area can be easily calculated by calculating the area of both the triangles separately.
Area of ABCD = Area of ∆ABC + Area of ∆ACD
= (1/2) × AC × h1 + (1/2) × AC× h2
The formula for the Area of a General Quadrilateral
Where h1 and h2 are the height of both the triangles.
d is the length of common diagonal i.e.AC.
Example
Find the area of quadrilateral ABCD.
Solution:
In the quadrilateral ABCD,
BD is the common diagonal so d = 5 cm.
Height of the two triangles are h1 = 2 cm and h2 = 1 cm.
Area of Special Quadrilaterals (Rhombus)
A rhombus is a quadrilateral with all the sides are equal and parallel but not the right angle. Its two diagonals are the perpendicular bisector to each other.
In this also we can split the rhombus into two triangles and can find the area of rhombus easily.
Formula of Area of Rhombus
Area of rhombus is half of the product of its two diagonals.
Area of a Polygon
There is no particular formula for the area of the polygon so we need to divide it in a possible number of figures like a triangle, rectangle, trapezium and so on. By adding the area of all the split figures, we will get the area of the required polygon.
Example
Find the area of the given octagon.
Solution:
We can divide the given octagon into three parts.
Two trapezium A and B and one rectangle shown by part B.
Two trapezium A and B and one rectangle shown by part B.
Area of A = Area of B = (1/2) × (a + b) × h
= (1/2) x (10 + 3) × 2
= 13 cm2.
Area of B = Length x Breadth
= 10 x 3
= 30 cm2.
So, the area of Octagon = 2A + B
= 2 × 13 + 30
= 56 cm2.