6. Parallelogram Formulas
A parallelogram is a quadrilateral in which opposite sides are parallel and equal.
In the diagram, let the two different side lengths be represented by a and b. The perpendicular height corresponding to base b is represented by h.
The important formulas for a parallelogram include area, perimeter and the relationship between its two diagonals.
What Is a Parallelogram?
b = Base | h = Height | a = Side | d₁, d₂ = Diagonals
The height h is the perpendicular distance between the base and its opposite parallel side. It is not necessarily the slanting side.
1. Area of a Parallelogram
The area of a parallelogram is calculated by multiplying its base by its corresponding perpendicular height.
If the base is b and height is h:
Example 1: Find the Area
A parallelogram has a base of 12 cm and a perpendicular height of 8 cm. Find its area.
A = 12 × 8
A = 96 cm²
Answer: 96 cm²
Example 2: Find Height from Area
The area of a parallelogram is 150 cm² and its base is 15 cm. Find its height.
From:
Therefore:
h = 150 ÷ 15
h = 10 cm
Answer: 10 cm
Example 3: Find Base from Area
The area of a parallelogram is 180 m² and its height is 12 m. Find its base.
b = 180 ÷ 12
b = 15 m
Answer: 15 m
2. Perimeter of a Parallelogram
A parallelogram has two sides of length a and two opposite sides of length b.
Therefore, the perimeter is:
Here:
- a = one side
- b = the other side
Example 1
The two sides of a parallelogram are 12 cm and 8 cm. Find its perimeter.
P = 2(12 + 8)
P = 2 × 20
P = 40 cm
Answer: 40 cm
Example 2: Find a Missing Side
The perimeter of a parallelogram is 60 cm. One side is 20 cm. Find the other side.
60 = 2(20 + b)
30 = 20 + b
b = 10 cm
Answer: 10 cm
3. Diagonal Relation of a Parallelogram
A parallelogram has two diagonals. They generally have different lengths.
Let the diagonals be d₁ and d₂, and the two sides be a and b.
The source formula gives the following important relationship:
What Do the Symbols Mean?
| Symbol | Meaning |
|---|---|
| a | First side of the parallelogram |
| b | Second side of the parallelogram |
| d₁ | First diagonal |
| d₂ | Second diagonal |
Example 1: Find the Sum of Squares of Diagonals
A parallelogram has sides 6 cm and 8 cm. Find the value of d₁² + d₂².
= 2(6² + 8²)
= 2(36 + 64)
= 2 × 100
= 200 cm²
Answer: 200 cm²
Example 2: Find One Diagonal
The sides of a parallelogram are 5 cm and 7 cm. One diagonal is 4√5 cm. Find the other diagonal.
Use:
Substitute:
d₁ = 4√5, a = 5, b = 7
Therefore:
(4√5)² + d₂² = 2(5² + 7²)
80 + d₂² = 2(25 + 49)
80 + d₂² = 148
d₂² = 68
Answer: 2√17 cm
4. Area and Perimeter — Do Not Confuse Them
| Quantity | Formula | Unit |
|---|---|---|
| Area | b × h | cm², m², etc. |
| Perimeter | 2(a + b) | cm, m, etc. |
| Diagonal Relation | d₁² + d₂² = 2(a² + b²) | Square units after squaring |
For area, use the perpendicular height. Do not automatically use the slanting side as the height.
For perimeter, use the actual side lengths.
5. Important Parallelogram Relationships
Area:
Therefore:
Perimeter:
Therefore:
Diagonal Relation:
6. Real-Life Example
Suppose a parallelogram-shaped piece of land has a base of 30 m and a perpendicular height of 20 m.
The area of the land is:
= 30 × 20
= 600 m²
So, the land covers an area of 600 square metres.
7. Common Mistakes in Parallelogram Questions
Mistake 1: Using the slanting side as height
The height must be the perpendicular distance between the parallel sides.
Mistake 2: Using the perimeter formula for area
Remember:
Mistake 3: Assuming the diagonals are always equal
Unlike a square or rectangle, the two diagonals of a general parallelogram are not necessarily equal.
Mistake 4: Forgetting the squares in the diagonal relation
The correct relation is:
8. Practice Questions
Question 1: Find the area of a parallelogram whose base is 15 cm and height is 8 cm.
Question 2: Find the perimeter of a parallelogram whose sides are 18 cm and 12 cm.
Question 3: The area of a parallelogram is 240 cm² and its base is 20 cm. Find its height.
Question 4: The area of a parallelogram is 360 m² and its height is 18 m. Find its base.
Question 5: The sides of a parallelogram are 6 cm and 8 cm. Find d₁² + d₂².
Question 6: The perimeter of a parallelogram is 80 cm and one side is 25 cm. Find the other side.
9. Parallelogram Formula Quick Revision
10. Mensuration Exam Tip
For quick revision, remember:
Area → Base × Perpendicular Height
Perimeter → 2(a + b)
Diagonals → d₁² + d₂² = 2(a² + b²)
Whenever a parallelogram diagram is given, first identify the base, perpendicular height, side lengths and diagonals. Then choose the appropriate formula.
In Part 7 — Quadrilateral Formulas, we will cover the important formulas for quadrilaterals with diagrams, explanations, solved examples, practice questions and competitive-exam tips.