How to Find the Perimeter of a Trapezium?
Last Updated :
24 Jul, 2024
Suppose the sides of a Trapezium are a, b, c, and d then the Perimeter of the Trapezium is (a + b + c + d) units.
Trapezium is a quadrilateral in which one pair of opposite sides are parallel. The perimeter of the trapezium is the sum of the boundaries of the trapezium. Suppose the sides of the trapezium are , b, c, and d then the perimeter of the trapezium is (a + b + c + d) units.
In this article, we will learn about What is Trapezium? Perimeter of Trapezium, Examples, and others in detail.
What is a Trapezium?
Trapezium is a quadrilateral containing one pair of opposite sides to are parallel. These parallel sets of sides are called bases and the non-parallel sides of a trapezium are called legs. It can be considered as a special case of a parallelogram with two parallel sides. The other name of Trapezium is Trapezoid.
Types of Trapezium
The trapezium can be categorized into three types that include,
- Isosceles Trapezium
- Scalene Trapezium
- Right Trapezium
Properties of Trapezium
Various Properties associated with a Trapezium are,
- One pair of opposite sides are parallel.
- Unequal non-parallel sides. Exception is the isosceles trapezium.
- The diagonals intersect each other.
- The line that intersects the mid-points of the non-parallel sides is always parallel to the bases. These are equal to half of the sum of parallel sides
Perimeter of Trapezium
The total summation of the length of the boundary of the trapezoid or trapezium is referred to as the Perimeter of Trapezium. Trapezium is an irregular polygon so we have no fixed formula to calculate the perimeter of trapezium. The unit of the perimeter of the trapezoid is the unit of length, such as "inches", "feet", "metres" etc.
Perimeter of Trapezium = Sum of Non-Parallel Sides + Sum of Parallel Sides
How to Find Perimeter of Trapezium?
The perimeter of Trapezoid can be found by following the steps added below,
Step 1: Note all the dimensions of the given Trapezium.
Step 2: Find the sum of all the dimensions of the trapezium.
Step 3: Put the unit of perimeter as the same unit as the length of the trapezium.
Let us assume a trapezoid ABCD, where we have sides AB and CD parallel to each other. They form the pair 'bases' while AD and BC are non-parallel sides (that is, they are legs). Let us assume the perimeter of the trapezoid is "P" units. The image of the assume trapezoid is added below,
Perimeter of trapezoid ABCD is given by,
P = AB + BC + CD + AD
Perimeter of Trapezoid in terms of Area is given by the formula,
P = (Sum of Lengths of Non-Parallel Sides) + 2 (A/h)
where,
- A is Area of Trapezoid
- h is Height
Read More:
Perimeter of Trapezium Examples
Example 1: Calculate the perimeter of a trapezium having parallel sides 4 cm and 6 cm. The non-parallel sides 5 cm and 8 cm?
Solution:
Given,
- Parallel sides are 4 cm and 6 cm
- Non-parallel sides are 5 cm and 8 cm
Perimeter of Trapezium is sum of all sides of Trapezium
Perimeter of Trapezium = Sum of Parallel Sides + Sum of Non-Parallel Sides
Perimeter of Trapezium = 4 cm + 6 cm + 5 cm + 8 cm
Perimeter of Trapezium = 23 cm
Example 2: Calculate the perimeter of a trapezium ground having parallel side 20 m 25 m and non-parallel side 18 m and 16 m?
Solution:
Given,
- Parallel sides are 20 m and 25 m
- Non-parallel sides are 18 m and 16 m
Perimeter of Trapezium is sum of all sides of Trapezium
Perimeter of Trapezium Ground = Sum of Parallel Sides + Sum of Non-Parallel Sides
Perimeter of Trapezium Ground = 20 m + 25 m + 18 m + 16 m
Perimeter of Trapezium Ground = 79 m
Example 3: If the perimeter of the trapezium is 54 cm and the parallel sides are 10 cm and 15 cm. Assume that the non-parallel sides are equal. Then find the length of both of the non-parallel sides?
Solution:
Given,
- Perimeter of Trapezium 54 cm
- Parallel sides are 10 cm and 15 cm
Non-parallel sides are equal, let assume non-parallel side be x
Therefore,
Perimeter of Trapezium = 54 cm
Perimeter of Trapezium = 10 cm + 15 cm + x + x
54 = 25 + 2x
2x = 54 - 25
2x = 29
x = 29/2 = 14.5 cm
Therefore the non-parallel sides of the trapezium are 14.5 cm and 14.5 cm.
Example 4: Calculate the perimeter of the regular trapezium having the parallel side 15 cm 17 cm, the length of the non-parallel side is 8 cm.
Solution:
Given,
- Parallel side of Trapezium are 15 cm and 17 cm
- Length of Non-Parallel sides are 8 cm and 8 cm
Perimeter of Trapezium = Sum of all Sides of Trapezium
Perimeter of Trapezium = 15 cm + 17 cm + 2(8)
= 15 cm + 17 cm + 16 cm
Perimeter of Trapezium = 48 cm
Example 5: Calculate the perimeter of a trapezium having parallel sides 25 cm and 30 cm. The non-parallel sides are 19 cm and 18 cm.
Solution:
Given,
- Parallel sides are 25 cm and 30 cm
- Non-parallel sides are 19 cm and 18 cm
Perimeter of Trapezium we need to add all the sides of the trapezium.
Perimeter of trapezium = Sum of Parallel Sides + Sum of Non-Parallel Sides
Perimeter of trapezium = 25 cm + 30 cm + 19 cm + 18 cm
Perimeter of trapezium = 92 cm
Example 6: A trapezium has parallel sides of 15 cm and 23 cm. If the non-parallel sides form angles of 60° and 45° with the longer parallel side, find the perimeter of the trapezium.
Solution:
Answer: The perimeter of the trapezium is approximately 51.25 cm.
Let's call the height of the trapezium h.
For the 60° angle: tan 60° = h / x, where x is the horizontal distance.
h = x * √3
For the 45° angle: tan 45° = h / y, where y is the horizontal distance.
h = y
The sum of x and y is the difference of parallel sides: x + y = 23 - 15 = 8
From the above equations: x√3 = y
x + x√3 = 8
x(1 + √3) = 8
x = 8 / (1 + √3) ≈ 3.31 cm
y = 8 - 3.31 = 4.69 cm
Non-parallel sides: √(3.31² + (3.31√3)²) ≈ 6.62 cm and √(4.69² + 4.69²) ≈ 6.63 cm
Perimeter = 15 + 23 + 6.62 + 6.63 ≈ 51.25 cm
Example 7 : The parallel sides of a trapezium are 18 cm and 30 cm. The height of the trapezium is 12 cm. If the area of the trapezium is 288 cm², calculate its perimeter.
Solution:
Answer: The perimeter of the trapezium is approximately 74.84 cm.
Area of trapezium = ½(a+b)h, where a and b are parallel sides and h is height
288 = ½(18+30) * 12
288 = 24 * 12 (This checks out, confirming our given information)
To find the non-parallel sides, we can use the Pythagorean theorem
Let x be half the difference of parallel sides: x = (30-18)/2 = 6 cm
Non-parallel side = √(x² + h²) = √(6² + 12²) = √180 ≈ 13.42 cm
Perimeter = 18 + 30 + 13.42 + 13.42 = 74.84 cm
Example 8 : In a trapezium ABCD, AB || DC, AB = 16 cm, DC = 24 cm. If the diagonals AC and BD intersect at point E such that AE:EC = 2:3 and BE:ED = 3:2, find the perimeter of the trapezium.
Solution:
Answer: The perimeter of the trapezium is approximately 57.88 cm.
The ratio of AE:EC = 2:3 implies that E divides AC in the ratio 2:3
Similarly, BE:ED = 3:2 implies that E divides BD in the ratio 3:2
In a trapezium, if diagonals are divided in these ratios, the non-parallel sides are equal
Let the non-parallel side be x
By the properties of trapeziums: AD/BC = AE/EC = 2/3
AD - BC = DC - AB = 24 - 16 = 8
Let BC = y, then AD = y + 8
y/y+8 = 2/3
3y = 2y + 16
y = 16, so AD = 24
Now we have a trapezium with sides 16, 24, x, x
We can find x using the Pythagorean theorem:
x² = 8² + ((24-16)/2)² = 64 + 16 = 80
x = √80 ≈ 8.94 cm
Perimeter = 16 + 24 + 8.94 + 8.94 = 57.88 cm
Example 9 : A trapezium has an area of 210 cm² and its parallel sides are in the ratio 4:3. If the height of the trapezium is 10 cm and one of the non-parallel sides is 13 cm, find the perimeter of the trapezium.
Solution :
Answer: The perimeter of the trapezium is approximately 31.42 cm.
Let the parallel sides be 4x and 3x
Area = ½(a+b)h = ½(4x+3x) * 10 = 210
35x = 42
x = 1.2
So parallel sides are 4.8 cm and 3.6 cm
To find the other non-parallel side, use Pythagorean theorem:
Let y be the other non-parallel side
y² = 10² + ((4.8-3.6)/2)² = 100 + 0.36 = 100.36
y ≈ 10.02 cm
Perimeter = 4.8 + 3.6 + 13 + 10.02 = 31.42 cm
Example 10 : The parallel sides of a trapezium are 20 cm and 28 cm. If the diagonals of the trapezium are perpendicular to each other and have lengths 26 cm and 30 cm, find the perimeter of the trapezium.
Solution:
Answer: The perimeter of the trapezium is 56 cm.
In a cyclic quadrilateral (which this trapezium is, due to perpendicular diagonals), we can use Ptolemy's theorem:
AC * BD = AB * DC + AD * BC
Let AB = 20, DC = 28, AC = 26, BD = 30
26 * 30 = 20 * 28 + AD * BC
780 = 560 + AD * BC
AD * BC = 220
We also know that AD + BC = 28 - 20 = 8
Using these two equations, we can solve:
AD + BC = 8 and AD * BC = 220
Let AD = x, then BC = 8 - x
x(8-x) = 220
8x - x² = 220
x² - 8x + 220 = 0
Solving this quadratic equation:
x ≈ 5.67 or 2.33
So AD ≈ 5.67 cm and BC ≈ 2.33 cm
Perimeter = 20 + 28 + 5.67 + 2.33 = 56 cm.
Practice Questions on Perimeter of Trapezium
Q1. Find the perimeter of the trapezium of sides, 12 units, 14 units, 20 units, and 16 units.
Q2. What is the perimeter of the trapezium of sides with sides 2 units, 4 units, 7 units, and 6 units?
Q3. What is the perimeter of Isosceles trapezium with equal sides 8 cm and other sides 9 cm and 7 cm?
Q4. If the perimeter of an Isosceles Trapezium is 60 cm and the unequal sides measures 22 and 14 cm then find the equal sides.
Q5 . Calculate the perimeter of a trapezium with parallel sides measuring 14 cm and 22 cm, and non-parallel sides of 9 cm each.
Q6.A trapezium has parallel sides of 18 cm and 26 cm. If the non-parallel sides form angles of 60° and 45° with the longer parallel side, find the perimeter of the trapezium.
Q7. The parallel sides of a trapezium are 15 cm and 25 cm. The height of the trapezium is 10 cm. If the area of the trapezium is 200 cm², calculate its perimeter.
Q8. In a trapezium ABCD, AB || DC, AB = 12 cm, DC = 20 cm. If the diagonals AC and BD intersect at point E such that AE:EC = 3:2 and BE:ED = 2:3, find the perimeter of the trapezium.
Q9. A trapezium has an area of 180 cm² and its parallel sides are in the ratio 5:3. If the height of the trapezium is 9 cm and one of the non-parallel sides is 11 cm, find the perimeter of the trapezium.
Q10. The parallel sides of a trapezium are 16 cm and 24 cm. If the diagonals of the trapezium are perpendicular to each other and have lengths 20 cm and 25 cm, find the perimeter of the trapezium.
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