Area of Parallelogram
A parallelogram is a four-sided shape (quadrilateral) where both pairs of opposite sides are parallel and equal in length. Examples of parallelograms include rectangles, rhombuses, and squares.
The area of a parallelogram tells us how much surface it covers. If you want to find how much cloth is needed for a parallelogram-shaped patch, or how much land a parallelogram-shaped field contains, you need to calculate its area.
In Class 7 Mathematics (NCERT), the area of a parallelogram is studied in the chapter Perimeter and Area. The formula is closely related to the area of a rectangle, and understanding this connection makes it easy to remember.
What is Area of Parallelogram?
Definition: The area of a parallelogram is the total space enclosed within its four sides. It tells us how much flat surface the shape covers.
What is a parallelogram?
- A parallelogram is a quadrilateral (4-sided shape) where both pairs of opposite sides are parallel and equal.
- Opposite angles are equal. Consecutive angles add up to 180°.
- Examples: rectangles, rhombuses, and squares are all special types of parallelograms.
Key terms:
- Base (b): Any one side of the parallelogram can be chosen as the base.
- Height (h): The perpendicular distance between the base and the opposite side. The height is NOT the slant side — it must be at right angles (90°) to the base.
- Opposite sides: In a parallelogram, opposite sides are parallel and equal.
- Slant side: The non-base sides that are tilted. These are NOT the height.
Important:
- The height is always measured perpendicular to the base, not along the slant side.
- If you choose a different side as the base, the corresponding height will also change.
- Area = base × corresponding height (the height that goes with that particular base).
- A very common mistake is using the slant side instead of the height — always draw a perpendicular line to find the height.
Area of Parallelogram Formula
Area of Parallelogram:
Area = base × height = b × h
Where:
- b = length of the base
- h = perpendicular height (the vertical distance between the two parallel sides)
Related formulas:
- To find base from area: b = Area / h
- To find height from area: h = Area / b
- Perimeter of parallelogram: P = 2(a + b), where a and b are the lengths of two adjacent sides.
Units:
- If the base and height are in cm, the area is in cm².
- If the base and height are in m, the area is in m².
Derivation and Proof
Deriving the formula from a rectangle:
Method (NCERT approach):
- Draw a parallelogram ABCD on paper.
- Drop a perpendicular from vertex D to the base AB. Call the foot of the perpendicular E. So DE = height (h).
- Cut the right triangle AED from the left side of the parallelogram.
- Move this triangle to the right side and attach it after side BC.
- The shape now becomes a rectangle with length = b (the base) and width = h (the height).
- Since the area has not changed by cutting and rearranging:
- Area of parallelogram = Area of rectangle = length × width = b × h.
Key insight:
- A parallelogram can always be rearranged into a rectangle with the same base and height.
- That is why Area of parallelogram = base × height, just like a rectangle.
- A rectangle is actually a special parallelogram where all angles are 90°.
Types and Properties
Types of area problems involving parallelograms:
1. Finding area given base and height:
- Directly apply Area = b × h.
- Example: base = 12 cm, height = 5 cm → Area = 12 × 5 = 60 cm².
2. Finding the missing dimension:
- If area and base are given, find height: h = Area/b.
- If area and height are given, find base: b = Area/h.
3. Comparing areas:
- Two parallelograms with the same base and same height have equal areas, regardless of the slant of the sides.
4. Area in different units:
- Convert units before calculating if base and height are in different units.
5. Cost-based problems:
- Find the area, then multiply by cost per unit area.
- Some shapes can be divided into parallelograms and other shapes.
Solved Examples
Example 1: Example 1: Finding area from base and height
Problem: Find the area of a parallelogram with base 15 cm and height 8 cm.
Solution:
Given:
- Base (b) = 15 cm
- Height (h) = 8 cm
Using the formula:
- Area = b × h
- Area = 15 × 8
- Area = 120 cm²
Answer: The area of the parallelogram is 120 cm².
Example 2: Example 2: Finding height from area and base
Problem: The area of a parallelogram is 180 cm² and its base is 20 cm. Find the height.
Solution:
Given:
- Area = 180 cm²
- Base (b) = 20 cm
Using the formula:
- Area = b × h
- 180 = 20 × h
- h = 180/20
- h = 9 cm
Answer: The height of the parallelogram is 9 cm.
Example 3: Example 3: Finding base from area and height
Problem: A parallelogram has an area of 252 m² and a height of 14 m. Find the base.
Solution:
Given:
- Area = 252 m²
- Height (h) = 14 m
Using the formula:
- Area = b × h
- 252 = b × 14
- b = 252/14
- b = 18 m
Answer: The base of the parallelogram is 18 m.
Example 4: Example 4: Area with decimal measurements
Problem: Find the area of a parallelogram with base 12.5 cm and height 6.4 cm.
Solution:
Given:
- Base (b) = 12.5 cm
- Height (h) = 6.4 cm
Using the formula:
- Area = b × h
- Area = 12.5 × 6.4
- Area = 80 cm²
Answer: The area is 80 cm².
Example 5: Example 5: Cost-based problem
Problem: A parallelogram-shaped garden has a base of 25 m and height 18 m. Find the cost of planting grass at Rs 40 per m².
Solution:
Given:
- Base = 25 m, Height = 18 m, Rate = Rs 40/m²
Step 1: Find the area:
- Area = b × h = 25 × 18 = 450 m²
Step 2: Find the cost:
- Cost = 450 × 40 = Rs 18,000
Answer: The cost of planting grass is Rs 18,000.
Example 6: Example 6: Comparing two parallelograms
Problem: Parallelogram A has base 10 cm and height 6 cm. Parallelogram B has base 12 cm and height 5 cm. Which has a greater area?
Solution:
Area of A:
- = 10 × 6 = 60 cm²
Area of B:
- = 12 × 5 = 60 cm²
Comparison:
- Both have the same area: 60 cm².
Answer: Both parallelograms have equal areas of 60 cm², even though they have different dimensions.
Example 7: Example 7: Unit conversion before calculation
Problem: A parallelogram has a base of 2 m and a height of 50 cm. Find the area in cm².
Solution:
Given:
- Base = 2 m = 200 cm (convert to cm)
- Height = 50 cm
Using the formula:
- Area = b × h = 200 × 50 = 10,000 cm²
In m²: 10,000 cm² = 1 m²
Answer: The area is 10,000 cm² or 1 m².
Example 8: Example 8: Using two different base-height pairs
Problem: A parallelogram ABCD has AB = 16 cm with corresponding height 9 cm, and AD = 12 cm. Find the height corresponding to AD.
Solution:
Step 1: Find the area using AB as base:
- Area = AB × h₁ = 16 × 9 = 144 cm²
Step 2: Use the same area with AD as base:
- Area = AD × h₂
- 144 = 12 × h₂
- h₂ = 144/12 = 12 cm
Answer: The height corresponding to side AD is 12 cm.
Example 9: Example 9: Area when height doubles
Problem: A parallelogram has base 10 cm and height 7 cm. If the height is doubled while the base stays the same, what happens to the area?
Solution:
Original area:
- = 10 × 7 = 70 cm²
New area (height doubled):
- New height = 2 × 7 = 14 cm
- New area = 10 × 14 = 140 cm²
Comparison:
- 140 = 2 × 70
- The area also doubles.
Answer: When the height is doubled, the area also doubles.
Example 10: Example 10: Parallelogram vs rectangle comparison
Problem: A rectangle and a parallelogram have the same base of 14 cm and same height of 10 cm. Compare their areas.
Solution:
Area of rectangle:
- = length × breadth = 14 × 10 = 140 cm²
Area of parallelogram:
- = base × height = 14 × 10 = 140 cm²
Comparison:
- Both have the same area.
Answer: A rectangle and a parallelogram with the same base and height have equal areas. This makes sense because a parallelogram can be rearranged into a rectangle.
Real-World Applications
Real-world uses of area of parallelogram:
- Land measurement: Many plots of land are parallelogram-shaped. Farmers and surveyors calculate the area to determine crop planting or property boundaries.
- Architecture: Some buildings, windows, and tiles use parallelogram shapes. Architects need to calculate areas for material estimation.
- Fabric and textile: Parallelogram-shaped patches in quilts, curtains, and clothing require area calculations for cutting fabric.
- Physics: The parallelogram law of vector addition uses parallelogram shapes to find the resultant of two forces.
- Road design: Some road signs and lane markings are parallelogram-shaped. Area calculations help determine paint needed.
- Floor tiling: Parallelogram-shaped tiles are used in patterns. The area of each tile helps calculate how many tiles are needed for a floor.
Key Points to Remember
- Area of parallelogram = base × height (b × h).
- The height is the perpendicular distance between the base and the opposite parallel side.
- The height is NOT the slant side of the parallelogram.
- Any side can be chosen as the base — but use the corresponding height for that base.
- A parallelogram with the same base and height as a rectangle has the same area.
- If the base is doubled (height same), the area doubles.
- If the height is doubled (base same), the area doubles.
- If both base and height are doubled, the area becomes 4 times.
- Area is always in square units (cm², m²).
- A rectangle is a special parallelogram where the height equals the breadth.
Practice Problems
- Find the area of a parallelogram with base 18 cm and height 11 cm.
- The area of a parallelogram is 324 m². If the base is 27 m, find the height.
- A parallelogram has base 22 cm and height 15 cm. Find the cost of painting it at Rs 5 per cm².
- A parallelogram PQRS has PQ = 20 cm with height 9 cm, and PS = 15 cm. Find the height corresponding to PS.
- The base of a parallelogram is three times its height. If the area is 192 cm², find the base and height.
- A rectangular field and a parallelogram-shaped field have the same base and height. If the rectangle has area 350 m², what is the area of the parallelogram?
- If the base of a parallelogram is halved and height is tripled, how does the area change?
- Find the area of a parallelogram with base 3.5 m and height 2.4 m.
Frequently Asked Questions
Q1. What is the formula for area of a parallelogram?
The area of a parallelogram = base × height (b × h). The base is any side, and the height is the perpendicular distance from the base to the opposite side.
Q2. Why is the height not the slant side?
The height must be perpendicular (at 90°) to the base. The slant side is not perpendicular to the base in most parallelograms. Using the slant side would give a wrong answer. Only in a rectangle is the side equal to the height, because all angles are 90°.
Q3. Is the area of a parallelogram the same as a rectangle with the same base and height?
Yes. A parallelogram can be cut and rearranged into a rectangle with the same base and height. So they have the same area. This is how the formula is derived.
Q4. Can any side be the base of a parallelogram?
Yes. You can choose any side as the base, but you must use the height that corresponds to that base (the perpendicular distance to the opposite parallel side). The area will be the same regardless of which side you choose.
Q5. What happens to the area if both base and height are doubled?
The area becomes 4 times the original. New area = (2b) × (2h) = 4bh = 4 × original area.
Q6. How is the area of a parallelogram different from the area of a triangle?
Area of parallelogram = b × h. Area of triangle = (1/2) × b × h. A triangle with the same base and height has exactly half the area of the parallelogram. A parallelogram can be split into two equal triangles by drawing a diagonal.
Q7. What are the units for area of a parallelogram?
Area is measured in square units. If the base and height are in cm, the area is in cm². If they are in metres, the area is in m². Always make sure both measurements are in the same unit before calculating.
Q8. Is a rhombus a parallelogram?
Yes. A rhombus is a special parallelogram where all four sides are equal. The area formula (base × height) still applies. Alternatively, for a rhombus, Area = (1/2) × d₁ × d₂, where d₁ and d₂ are the diagonals.
