How to Find the Area of a Parallelogram: Step-by-Step with Formulas and Tricks
The concept of area of parallelogram plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding how to find the area of a parallelogram helps students master geometry, solve word problems, and compare different quadrilaterals efficiently.
What Is Area of Parallelogram?
A parallelogram is a four-sided, closed 2D shape (quadrilateral) with both pairs of opposite sides parallel. The area of parallelogram is the space covered within its boundaries. You’ll find this concept applied in areas such as area of triangle, land measurement, and computer graphics.
Key Formula for Area of Parallelogram
Here’s the standard formula: Area = base × height
\( \text{Area} = b \times h \)
where b is the length of the base and h is the perpendicular height (altitude) from the base to the opposite side.
Alternative Formulas for Area of Parallelogram
In some questions, base and height may not be given directly. You can also calculate the area using:
- Area = a × b × sin(θ): Where a and b are adjacent sides, and θ is the included angle.
- Area = ½ × d₁ × d₂ × sin(α): Where d₁ and d₂ are diagonals, and α is their intersection angle.
- Area = |A × B|: Using vector cross product when sides are given as vectors.
Cross-Disciplinary Usage
Area of parallelogram is not only useful in Maths but also plays an important role in Physics (calculating work and torque), Computer Science (graphics, collision detection), and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions and competitive exams.
Step-by-Step Illustration
- Suppose the base of a parallelogram is 10 cm and the height is 6 cm.
- Use the area of parallelogram formula:
Area = base × height - Substitute the values:
Area = 10 × 6 = 60 cm2 - So, the area of parallelogram is 60 cm2.
Examples Using Different Formulas
| Given | Formula Used | Working | Answer |
|---|---|---|---|
| Base = 8 cm, Height = 4 cm | Area = b × h | 8 × 4 | 32 cm² |
| a = 7 m, b = 5 m, θ = 60° | Area = a × b × sinθ | 7 × 5 × sin60° = 35 × 0.866 | 30.31 m² |
| d₁ = 9 cm, d₂ = 12 cm, α = 30° | Area = ½ × d₁ × d₂ × sinα | 0.5 × 9 × 12 × 0.5 | 27 cm² |
| Vectors A = 3i + j, B = i + 2j | Area = |A × B| | |3i + j × i + 2j| = |(3×2)-(1×1)| = |6-1| | 5 units² |
Speed Trick or Vedic Shortcut
Here’s a quick shortcut to spot the area of parallelogram fast when you know only the sides and included angle:
- Multiply both side lengths (a × b)
- For easy angles like 30°, 60°, or 90°, remember common values: sin30° = 0.5, sin60° = 0.866, sin90° = 1
- Directly substitute to speed up calculations!
For more tricks and exam tips, try Vedantu's live classes where teachers explain how to avoid calculation mistakes while finding area or perimeter of parallelogram.
Try These Yourself
- Find the area of a parallelogram with base 15 cm and height 9 cm.
- If two adjacent sides are 12 m and 8 m and included angle is 90°, what is the area?
- Given diagonals 10 cm, 10 cm, and intersection angle 60°, find area.
- If area is 120 m² and base is 10 m, what is the height?
Frequent Errors and Misunderstandings
- Confusing base with side—not using the perpendicular height.
- Using the wrong angle (must be between adjacent sides or diagonals).
- Mistaking parallelogram’s area formula for rectangle or triangle.
- Missing units (always write cm², m², etc.).
Relation to Other Concepts
The idea of area of parallelogram connects closely with topics such as area of rectangle, area of rhombus, and area of a quadrilateral. Mastering this helps you solve coordinate geometry problems and understand special shapes like rectangles, squares, and rhombuses.
Classroom Tip
A quick way to remember area of parallelogram is to imagine “base times height always meets at a right angle.” Drawing a dotted line from a vertex to the base as the height helps visualize this. Vedantu’s teachers often show this trick in class to ensure you always pick the perpendicular height, not the slant one.
We explored area of parallelogram—from simple definition, standard and alternative formulas, worked examples, frequent mistakes, and its links to other shapes. Continue practicing problems and checking your answers with the Area of Parallelogram Calculator, and join Vedantu for live sessions to boost your math confidence!
Area of Triangle | Area of Rectangle | Area of Rhombus | Area of Parallelogram Calculator
FAQs on Area of Parallelogram – Formula, Derivation, and Examples
1. What is the formula for the area of a parallelogram?
The area of a parallelogram is calculated by multiplying its base (b) by its perpendicular height (h). The formula is: Area = b × h square units. The height is the perpendicular distance between the base and the opposite side.
2. How do I find the area of a parallelogram if the height isn't given?
If the height isn't given, you can use trigonometry. If you know the lengths of two adjacent sides (a and b) and the angle (θ) between them, the area is: Area = a × b × sin(θ) square units. Alternatively, if you know the lengths of the diagonals (d1 and d2) and the angle (φ) between them, the area is: Area = ½ × d1 × d2 × sin(φ) square units.
3. What is the difference between the area of a parallelogram and a rectangle?
A rectangle is a special type of parallelogram where all angles are 90 degrees. The area formula is the same (base × height), but in a rectangle, any side can be considered the base, and the adjacent side is the height. A parallelogram's height must be the perpendicular distance between the base and opposite side.
4. How do I find the area of a parallelogram using vectors?
If you have the vectors representing two adjacent sides of the parallelogram, say vector a and vector b, the area is given by the magnitude of their cross product: Area = |a × b| square units. This is equivalent to finding the magnitude of the vector which is perpendicular to the plane formed by the vectors a and b.
5. What are the units for the area of a parallelogram?
The units for area are always square units. For example, square centimeters (cm²), square meters (m²), or square kilometers (km²), depending on the units used for the base and height.
6. Can a parallelogram have a negative area?
No, the area of a parallelogram is always a positive value. While calculations involving vectors might yield a negative sign, it only indicates the orientation of the vectors and not a negative area.
7. How does measurement error affect the calculated area of a parallelogram?
Errors in measuring the base or height directly affect the calculated area. A small error in either measurement will lead to a proportionally larger error in the calculated area. Accurate measurements are crucial for a reliable result.
8. What are some real-world applications where calculating the area of a parallelogram is useful?
Calculating the area of a parallelogram is useful in many real-world situations, including: determining the area of a plot of land (often simplified as a parallelogram), calculating the amount of material needed to cover a surface, and in various engineering and architectural designs.
9. How is the area of a parallelogram related to the area of a triangle?
A parallelogram can be divided into two congruent triangles by drawing a diagonal. Therefore, the area of a parallelogram is twice the area of one of those triangles. This relationship is often useful in problem-solving.
10. What common mistakes do students make when calculating the area of a parallelogram?
Common mistakes include: using the wrong formula (confusing it with the area of a rectangle, rhombus, or triangle); forgetting that the height must be perpendicular to the base; and using incorrect units or not squaring the units for the area.
11. What is the relationship between the area of a parallelogram and its diagonals?
The area of a parallelogram can be calculated from its diagonals if the angle between them is known. The formula in this case is: Area = 1/2 * d1 * d2 * sin(θ), where d1 and d2 are the lengths of the diagonals, and θ is the angle between them.
12. How does the area of a parallelogram change if we change its base or height?
The area of a parallelogram is directly proportional to both its base and its height. If you double the base, the area doubles. If you double the height, the area doubles. If you double both, the area quadruples.
