The Area of Trapezium is the measure of the space enclosed within its boundaries. A
trapezium is a four-sided polygon
,
or quadrilateral, with two parallel sides of unequal lengths and two non-parallel sides that can vary in length and angle.
The two parallel sides are known as the "bases," and the perpendicular distance between them is the "height" or "altitude."
To calculate the area of trapezium, a specific formula is used:
12×(𝑎+𝑏)×ℎ21
, where 𝑎and 𝑏 represent the lengths of the parallel sides, and ℎ is the height.
Understanding this formula allows for accurate measurement of the trapezium's area, which is essential in geometry, architecture, and various real-life applications.
What is Area of Trapezium?
The area of trapezium is determined using the formula
12×(sum of its parallel sides)×(height)21×(sum of its parallel sides)×(height)
. This can be simplified by first calculating the average length of the parallel sides, which then results in the formula
(average length)×height (average length)×height
. Here "height" refers to the perpendicular distance between the two parallel sides of the trapezium.
Therefore, the area of trapezium depends on the lengths of its parallel sides and the distance (height) between them.
Also Check:
Square Root
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Properties of Trapezium
A trapezium (also known as a trapezoid in American English) is a four-sided polygon with one pair of parallel sides. Below are the key properties of a trapezium:
-
Two Parallel Sides
: A trapezium has exactly one pair of parallel opposite sides. These sides are called the bases of the trapezium, typically labelled as 𝑎 and 𝑏.
-
Non-parallel Sides
: The other two sides of a trapezium are called the legs. These legs are not parallel to each other. Their lengths can vary.
-
Height: The perpendicular distance between the two parallel sides (the bases) is called the height (or altitude) of the trapezium. It is the shortest distance between the two bases.
-
Angles
: The adjacent angles between the parallel sides are supplementary, meaning they add up to 180°. Specifically, one pair of consecutive angles (one on each side of the trapezium) is supplementary.
-
Symmetry
: Unlike other quadrilaterals, a trapezium typically does not have any line of symmetry unless it is an isosceles trapezium (where the legs are equal in length).
-
Diagonals
: The diagonals of a trapezium are generally not equal in length and do not bisect each other at right angles. However, in an isosceles trapezium, the diagonals are equal in length.
Area
: The area of trapezium is given by the formula:
Area=12×(𝑎+𝑏)×ℎ
Where
𝑎 and 𝑏 are the lengths of the parallel sides, and
ℎ is the height.
Perimeter
:
The perimeter of a trapezium is the sum of the lengths of all four sides.
Perimeter=𝑎+𝑏+leg1+leg2
Where
𝑎 and 𝑏 are the lengths of the parallel sides, and
Leg 1 and leg 2 are the lengths of the non-parallel sides.
Isosceles Trapezium: A special type of trapezium is the isosceles trapezium, where the non-parallel sides (legs) are equal in length. In this case:
-
The angles at each base are equal.
-
The diagonals are equal in length.
-
The line of symmetry divides the trapezium into two mirror-image parts.
Circumscribed Circle
: A trapezium generally does not have a circumscribed circle, except for an isosceles trapezium, which can have a circle inscribed inside it (where the circle touches all four sides).
Also Check:
Trapezium
Derivation of Area of Trapezium
To derive the area of trapezium, consider a trapezium with two parallel sides of lengths 𝑎 and 𝑏 and height ℎ, which is the perpendicular distance between these two parallel sides. To begin the derivation, imagine splitting one of the legs of the trapezium into two equal parts.
Then, cut a triangular portion from the trapezium and move it to the bottom, rearranging the shape. This rearrangement transforms the trapezium into a triangle where the base is now the sum of the two parallel sides, 𝑎+𝑏, and the height remains the same, ℎ.
The formula 12× Base × Height gives the area of a triangle, and in this case, the base of the triangle is 𝑎+𝑏, and the height is ℎ
Therefore, the area of the trapezium is equal to the area of the triangle, which results in the formula:
12×(𝑎+𝑏)×ℎ21 ×(a+b)×h. This
is the standard formula used to calculate the area of trapezium.
How to Calculate Area of Trapezium?
To calculate the area of trapezium, you need to know the lengths of its parallel sides and the height. The
parallel sides are typically labeled as 𝑎 (top base) and 𝑏(bottom base), while the height ℎ is the perpendicular distance between these two sides
. Once you have these measurements, you can apply a straightforward formula to find the area of the trapezium.
-
Identify the Dimensions
:
First, gather the necessary measurements of the trapezium. You need to find the lengths of the two parallel sides, which are often referred to as the "bases" of the trapezium. Let’s call them a (the top base) and b (the bottom base). Additionally, you need the height of the trapezium, denoted as h, which is the perpendicular distance between these two parallel sides.
-
Add the Lengths of the Parallel Sides
:
Once you have the lengths of the two parallel sides, add them together. This gives you the combined length of both the top and bottom bases of the trapezium.
-
Multiply by the Height
:
After summing the lengths of the parallel sides, multiply this total by the height of the trapezium. The height is the straight-line distance between the two parallel sides. This step gives you the area of a rectangle that would span the entire width of the trapezium.
-
Apply the Factor of One-Half
:
Since a trapezium is not a full rectangle but rather a quadrilateral with slanted sides, the area is only half of the rectangle's area. Therefore, multiply the result from the previous step by 1/1//2.2*1. This will give you the final area of the trapezium.
Also Check:
Isosceles Triangle
Area of Trapezium Derivation Using a Parallelogram
To understand the area of trapezium, one helpful approach is to derive it using a parallelogram. This method simplifies the process by comparing the trapezium to a parallelogram and using the properties of both shapes.
Step-by-Step Derivation
:
Consider the Trapezium
: Let the trapezium have two parallel sides, 𝑎 and 𝑏 (where 𝑎 is the top base and 𝑏 is the bottom base). The height of the trapezium is denoted as ℎ, which is the perpendicular distance between the two parallel sides.
Create a Parallelogram
: I
magine transforming the trapezium into a parallelogram. One way to do this is to replicate the trapezium and rotate one part of it. You can form a parallelogram by sliding the top portion of the trapezium horizontally and attaching it to the bottom. In this new shape, the combined length of the parallel sides 𝑎 and 𝑏 becomes the base of the parallelogram, and the height remains the same as ℎ
Area of the Parallelogram
: The area of a parallelogram is calculated using the formula:
Area of Parallelogram=Base×Height
In this case, the base is
𝑎+𝑏 (the sum of the parallel sides), and the height remains ℎ
So, the area of the parallelogram is:
Area of Parallelogram=(𝑎+𝑏)×ℎ
Relating the Trapezium and Parallelogram
:
The original trapezium is half of the newly formed parallelogram. This is because when you replicate and slide the top portion of the trapezium, you are essentially doubling the space occupied by the trapezium. So, the area of the trapezium is half the area of the parallelogram.
Final Formula
:
Since the trapezium is half the parallelogram, the area of the trapezium is:
Area of Trapezium=12×(𝑎+𝑏)×ℎ
This is the standard formula for calculating the area of trapezium.
Area of Trapezium by Heron's Formula
Heron’s formula is typically used to find the area of a triangle when the lengths of all three sides are known. However, we can extend its use to find the area of a trapezium by dividing it into two triangles.
-
Step-by-Step Derivation
:
Consider a trapezium
ABCD, where
AB∥CD are the parallel sides, and the other two sides are AD and BC.
-
The height h is the perpendicular distance between the parallel sides
AB and CD.
-
To use
Heron’s formula for a trapezium, we need to divide the trapezium into two triangles by drawing a height from one vertex to the opposite base.
Let's assume the trapezium is divided into two triangles by drawing a perpendicular height from point
𝐶 to base
AB, and call the point of intersection 𝐸
The trapezium is now
divided into two triangles: △ABE and △CDE.
1. Divide the Trapezium into Two Triangles
Let:
a=AB (the length of one base)
b=CD (the length of the other base)
c=AD (side of the trapezium)
d=BC (side of the trapezium)
ℎ (height, which is the perpendicular distance between the two bases)
2. Calculate the Area of Each Triangle
For each triangle formed by dividing the trapezium, we can apply Heron's formula.
Triangle 1
:
△ABE
The sides of triangle
ABE are:
AB=a
BE=h
AE= the distance between point
𝐴
A and the foot of the perpendicular from
C (we will calculate this distance next).
The area of triangle
△ABE using Heron’s formula is: 𝐴1=𝑠1(𝑠1−𝑎)(𝑠1−ℎ)(𝑠1−𝐴𝐸)
Triangle 2
:
△CDE
Similarly, apply Heron's formula to triangle
CDE with:𝐴2=𝑠2(𝑠2−𝑏)(𝑠2−ℎ)(𝑠2−𝐵𝐸)
3. Calculate the Area of the Trapezium
Add the areas of the two triangles to get the area of the trapezium:
Area of Trapezium=𝐴1+𝐴2
This method extends the use of Heron's formula, but it's a more complex approach, often requiring additional geometric steps (like calculating segment lengths and solving for the height). In practice, it's more efficient to use the standard trapezium area formula:
Area of Trapezium=12×(𝑎+𝑏)×ℎ
Where
:
𝑎 and 𝑏 are the lengths of the parallel sides.
ℎ is the height (perpendicular distance between the parallel sides).
However, Heron's formula could be applied indirectly for certain geometric conditions when you have access to the side lengths, making it useful for breaking down complex shapes like a trapezium.
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Area of Trapezium Derivation Using a Triangle
To derive the area formula for a trapezium, let's follow a step-by-step approach using a triangle. Consider a trapezium with bases aaa and bbb and height h. Here's the process:
Step 1
: Split one of the legs of the trapezium into two equal parts.
Step 2
: Cut a triangular portion from the trapezium, as shown in the diagram.
Step 3
: Attach this triangular portion to the bottom of the trapezium, as shown in the revised diagram.
Proof of Area of Trapezium Formula:
After rearranging, the shape now forms a triangle. It can be concluded that the areas of both the trapezium and the triangle are equal. The base of the triangle is now
𝑎+𝑏, and its height remains ℎ
Therefore, the area of the trapezium is equal to the area of the triangle, which can be calculated as:
Area of Trapezium=Area of Triangle=12×(Base)×(Height)=12×(𝑎+𝑏)×ℎ
This is the derived formula for the area of trapezium.
