Area and perimeter are the fundamental topics in the mathematics as helps in providing foundation for the advanced mathematics and its applications. There is a practical application of the knowledge of the area and perimeters and it is very much required by the general people. Areas and perimeter of any figures helps people in gaining an understanding of the mathematic concepts and its application. The discipline of calculus requires the use of trapezoid. This is because, areas under the curve is determined by using calculus. Areas under the curve can be approximated by a series of trapezia that is one side that is slant to the approximate of the slope, two sides that are running parallel to the y axis and one side along the x axis.
A trapezium is a type of quadrilateral having four sides out of which, two sides are parallel and it has the common property that all the sum of angles would be equal to 3600. The diagonals, angles and side of trapezium are congruent. The area of trapezium is derived by calculating the area by dividing it into three sections that is two triangles and a rectangle. Trapezium is also known by other name that is trapezoid which is a type of polygon. Therefore, it is a quadrilateral which has one pair having parallel sides that are opposite. The two non-parallel sides of the trapezium are called legs and the two parallel sides are known as bases. Trapezium is known as isosceles trapezium if they form an equal angles at one of the bases and if the two sides that are non-parallel are equal. In addition to this, the mid-point of the non-parallel side of trapezoid is connected by the line segment known as the mid segment. The trapezium would be divided into two unequal parts if the line segment from the centre of both the sides is drawn between the two parallel lines. Therefore, it can be inferred that like all the other quadrilaterals, a trapezium has four sides that are unequal, sum of all the angles being equivalent to 3600 and the diagonals bisecting each other. Furthermore, there is no congruency between the angles, sides and diagonals of the trapezium.
The above figure represents a trapezium which has been divided into two triangles and one rectangle. The base of both the triangles combined is the difference between the two parallel sides length. That is the base of the triangles combined is of the length (m-l). This can be represented separately in the diagram below. The area of trapezium is equivalent to the sum of area of rectangle and areas of sum of two triangles.
The area of trapezium will be the sum of area of the triangle and rectangle above that is it can be written in the form of formula that is
Area of Trapezium= Area of triangle + Area of rectangle
It is well known that the area of triangle is obtained by using the formula where base and height of a triangle is multiplied by half. So, area of triangle is equals to [1/2*h*(m-l)].
Now, area of rectangle is computed by multiplying length by the breadth that is h*l.
Therefore, area of trapezium is hl + 1/2*h*(m-l).
Area of trapezium= h [l + ½(m-l)]
= h [2l/2 + m-l/2]
= h [2l + m- l/2]
= Distance between two parallel lines * (half of the sum of parallel sides)
The area of trapezium can be depicted using one of the examples.
The area of trapezium can be found when the length of the sides of the trapezium ABCD is given that is AB and CD is given.
BD= 13 cm
The value of PD can be obtained by the difference between CD and CP, that is (25-20) = 5 cm.
Now, the area of trapezium is computed by adding up the area of triangle BPD and area of square ACPB.
Firstly, the length of BP is obtained by using the Pythagoras theorem that is
BD= √ (BP2 + PD2)
BP= √ (BD2 - PD2) = √169-25= √144= 12 cm
Lastly, the area of trapezium is computed by adding up the area of triangle and area of square.
Area of trapezium ACDB= AB * BP + ½ (PB * PD) = (20 * 12 + ½ * 12* 5) = 240 + 30= 270 cm.
Therefore, the area of trapezium is 270 cm2.
Perimeter is the path that surrounding a two dimensional shape and it represented by the continuous lines that forms the boundary of the closed figure. Therefore, the perimeter of a trapezium is the sum of the length of all the four sides. If the length of all the sides of trapezium us given, it is quiet easy to compute the value of the perimeter of the trapezium. In the event, when one or more lengths of the trapezium is unknown or is not provided, then the length of that particular sides, can be ascertained using the Pythagoras theorem.
This can be explained with the help of an example.
If the length of three sides that is AB, CD and BD is given and AC is unknown.
Say, CD= 10 cm
BD= 8 cm
AB= 7 cm
AE= 5 cm
CE= CD-AB= 10 – 7= 3 cm.
Length of AC can be found using Pythagoras theorem,
AC= √ (AE2 + CE2)
= √25+9= √35= 5.91
Therefore, the perimeter of trapezium ABCD is the sum of all the four sides, that is
Area of trapezium = AB + BD + CD +AC= (7 + 8 + 5.91 + 10) = 30.91 cm
The concept of trapezium has a wide range application and it is highly used in several other mathematical calculations and physical computations. The concept of trapezium forms the basis of obtaining the motion equation and this is very essential to clear the level of understanding of the engineering mind
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