Trapezoid - Definition, Types, Properties and Formulas
Last Updated : 17 Sep, 2024
Trapezoid is another name for trapezium. It is a quadrilateral in which one pair of opposite sides are parallel. These parallel sides are known as the bases of the trapezoid, and the non-parallel sides are called the legs. It is a quadrilateral and follows all the properties of a quadrilateral.
In this article, we will discuss trapezoids, their definition, types, properties of a trapezoid, formulas of a trapezoid, and related examples along with the practical uses of trapezoids in real life.
What is a Trapezoid?
Trapezoid is a quadrilateral that has one pair of parallel sides and another set of non-parallel sides. The parallel sides are known as bases. The other two non-parallel sides are called lateral sides. Altitude or height is the shortest distance between the two parallel sides.
A trapezoid is shown in the image below:
Types of Trapezoid
There are three types of trapezoids that are:
- Isosceles Trapezoid: This type of trapezoid have equal-length legs ( non parallel sides) and equal angles opposite these legs. Isosceles trapezoids are symmetrical, which allows for easier calculation of area and other properties.
- Right-Angled Trapezoid: It is a type of trapezoid where at least two adjacent angles are right angles.
- Scalene Trapezoid: In this type of trapezoid , the legs are of unequal length, and none of the non-adjacent angles are equal. These trapezoids lack the symmetrical properties.
Properties of a Trapezoid
Various properties of a Trapezoid are:
- They has only one pair of parallel sides.
- Sum of all angles of a Trapezoid is 360o.
- For a Isosceles Trapezoid, both base angles are congruent , non parallel sides are congruent and diagonals are congruent.
- Median is parallel to both the bases and is the average length of the bases in an trapezoid.
- Adjacent angles ( one obtuse and one acute) of a trapezoid sum to 180o.
Area of Trapezoid
Let us consider a trapezoid (trapezium) PQRS, SR and PQ are parallel sides and ST is the shortest distance between them which is perpendicular and is denoted with h.
Let‘s divide the trapezoid into segments. That gives us two triangles PTS and UQR and a rectangle TURS. Now if we find the areas of all three and sum them, we get the area of the Trapezoid.
Area of triangle, PTS = 1/2 (base × height)
Area of triangle, PTS = 1/2 (PT × ST)
= 1/2 (PT*h)......(1)
On writing the area of rectangle TURS = length × breadth
Area of rectangle TURS = TU*h......(2)
In writing the area of triangle UQR = 1/2 (base × height)
Area of triangle UQR = 1/2 ( UQ*h).....(3)
In summing up the three areas i.e. equations (1), (2), and (3), we get the area of the trapezium.
On adding all the three equations,
Area of Trapezoid = 1/2 (PT*h) + TU*h + 1/2 ( UQ*h)
on taking 1/2 (h) common, we can write the equation as
Area of Trapezoid = (1/2) h (PT + 2TU + UQ)
Area of Trapezoid = (1/2) h (PT + TU + TU + UQ)
PT + TU + UQ = PQ and TU = SR
Area of Trapezoid = (1/2) × h × (PQ+SR)
From the figure, we can observe that summing x, b, and y gives a. So,
Area of Trapezoid = (1/2) × h × (a+b)
or
Area of Trapezoid = (1/2) × (height) × (sum of the parallel sides)
Perimeter of Trapezoid
For finding the perimeter, the lengths of all the sides are to be added.
Perimeter of Trapezoid = Sum of lengths of parallel and non-parallel sides
For any trapezoid ABCD with sides, AB = a, BC = b, CD = c, DA = a, its perimeter is:
Perimeter of Trapezoid = a + b + c + d
Hence, the Perimeter of a trapezoid is the sum of lengths of parallel sides and the sum of lengths of non-parallel sides.
Uses of Trapezoid/Trapezium in Real Life
- Architecture and Construction: Trapezoids provide aesthetic appeal and structural stability in buildings and bridges.
- Civil Engineering: They are essential in designing efficient water drainage systems for roads and channels.
- Furniture Design: Trapezoidal shapes are used in contemporary furniture designs for both stability and style.
- Graphic Design: They help create a sense of depth and perspective, enhancing visual compositions in graphic projects.
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Example 1: Find the area of trapezoid if the bases are 10 cm and 16 cm. The shortest distance between the parallel sides is 8 cm.
Solution:
Given lengths of bases,
- a = 10 cm
- b = 16cm
- height h = 8cm
Area of Trapezoid = (1/2) h (a+b)
= (1/2) 8 (10+16)
= 140 cm2
Example 2: If the area of a trapezoid is given as 240 cm2 and the sum of lengths of parallel sides is given as 30 cm, find the height of the trapezoid.
Solution:
Given,
- Area = 240 cm2
- (a+b) = 30 cm
Area of Trapezoid = (1/2) h (a+b)
240 = (1/2) h (30)
h = (240 × 2) / 30
h = 16 cm
Hence, the height of trapezoid is 16 cm
Example 3: Find perimeter of trapezoid if lengths of the sides are 15cm, 6cm, 12 cm, and 8 cm respectively.
Solution:
Given,
- a = 15 cm
- b = 6 cm
- c = 12 cm
- d = 8 cm
Perimeter of Trapezoid = Sum of lengths of parallel sides and sum of lengths of non parallel sides
P = a + b + c + d
P = 15+6+12+8
P = 41 cm
Example 4: If the perimeter of a trapezium is 48 cm and the sum of parallel sides is 26 cm and the length of the Third side is 10 cm. Find the length of the fourth side.
Solution:
Given,
- a+b = 26 cm
- c =10 cm
- Perimeter = 48 cm
Perimeter of Trapezoid = Sum of lengths of parallel sides and the sum of lengths of nonparallel sides
Perimeter of Trapezoid = a + b + c + d
48 = 26 + 10 + d
d = 48- 26 - 10
= 12 cm
Hence the length of the fourth side is 12 cm
Example 5: If the length of a parallel side is greater than the other by 6 cm and the area of a trapezoid is 240 cm2. Find the lengths of the parallel sides If the shortest distance between the parallel sides is 14 cm.
Solution:
Given,
- Area = 240 cm2
- Height = 14 cm
Let a parallel side a = x
According to given condition
b = x + 6
Area of trapezium = (1/2) h (a+b)
240 = (1/2) (14) (x + x + 6)
(240×2) / 14 = 2x + 6
2x = 34.28 - 6
2x = 28.28
x = 14.14
Hence a = 14.14 cm
b = x + 6
b = 14.14 + 6
b = 20.14 cm
Hence 14.14 cm and 20.14 cm are the lengths of the parallel sides
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