How to Find the Area and Perimeter of a Rectangle with Examples
The concept of rectangle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From classroom geometry to daily objects like books, screens, and tables, the rectangle is one of the most recognized shapes. Let’s understand its definition, properties, formulas, and how to solve rectangle problems easily.
What Is Rectangle?
A rectangle is a four-sided polygon (quadrilateral) with opposite sides equal and all four interior angles exactly 90 degrees. You’ll find this concept applied in measurement, geometry problems, and practical design scenarios. Many basic objects like an A4 sheet, blackboard, TV screen, and mobile phone are in the shape of a rectangle, making it an important foundation in maths education.
Key Formula for Rectangle
Here are the standard formulas for a rectangle:
- Area of Rectangle: \( A = L \times W \) (where L = length, W = width or breadth)
- Perimeter of Rectangle: \( P = 2 (L + W) \)
- Diagonal of Rectangle: \( D = \sqrt{L^2 + W^2} \)
Properties of Rectangle
- All four angles are right angles (each is 90°).
- Opposite sides are equal and parallel.
- Both diagonals are equal in length.
- The diagonals bisect each other (they cross at the center).
- A rectangle is a special parallelogram with all angles equal.
- The sum of the interior angles is always 360°.
- A rectangle has two lines of symmetry.
Difference Between Rectangle and Square
| Rectangle | Square |
|---|---|
| Opposite sides equal, adjacent sides can differ | All four sides equal in length |
| All angles are 90° | All angles are 90° |
| Diagonals equal and bisect each other | Diagonals equal, bisect each other at right angle (90°) |
| 2 lines of symmetry | 4 lines of symmetry |
Step-by-Step Illustration
Let’s solve a typical rectangle problem:
Question: The sides of a rectangle are 5 cm and 12 cm. Find the area, perimeter, and length of the diagonals.
1. Given: Length (L) = 5 cm, Width (W) = 12 cm2. Area: \( A = L \times W = 5 \times 12 = 60\ \mathrm{cm}^2 \)
3. Perimeter: \( P = 2(L + W) = 2(5 + 12) = 2 \times 17 = 34\ \mathrm{cm} \)
4. Diagonal: \( D = \sqrt{L^2 + W^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\ \mathrm{cm} \)
Final Answers: Area = 60 cm², Perimeter = 34 cm, Diagonal = 13 cm.
Speed Trick or Vedic Shortcut
A rapid way to find if a quadrilateral is a rectangle is to check if it has one right angle and its diagonals are equal. You don’t need to check all angles every time in competitive exams. Remember, for unknown side problems, rearrange the area or perimeter formula to solve quickly!
Example Trick: If area and length are given, you can instantly find width by \( W = \frac{A}{L} \).
Vedantu teachers often share more such tips to boost your maths speed and accuracy.
Try These Yourself
- Find the area and perimeter of a rectangle with length 7 cm and width 9 cm.
- If the diagonal is 10 cm and length is 6 cm, what is the width?
- List three daily-life objects that are rectangles.
- Explain why a square is a type of rectangle.
Frequent Errors and Misunderstandings
- Mixing up length and width (always label clearly).
- Using wrong units (area is in square units, perimeter is in regular units).
- Forgetting to double while calculating perimeter (2 × (L + W)).
- Assuming a rectangle and parallelogram are always the same (but only rectangles have all right angles).
Relation to Other Concepts
The idea of rectangle connects closely with quadrilaterals and parallelograms. Mastering rectangles helps with understanding area, perimeter, symmetry, and other two-dimensional shapes in further chapters.
Classroom Tip
A quick way to remember rectangle properties: “Opposite sides are equal, all angles are right, and diagonals cross at the center.” Drawing and labeling rectangles regularly in your notebook will help fix the formulas in your mind. Vedantu’s live classes often use practical objects like books and screens to make this more visual for students.
Real-Life Examples of Rectangle
- Book covers
- Mobile phone screens
- Doors and windows
- A4 size sheets and envelopes
- Television and monitor displays
Rectangle in Other Subjects
Rectangle is not only useful in Maths but also vital in Physics (area calculations), Computer Science (grid layouts), Engineering and daily logic puzzles. Students preparing for JEE, NEET, and Olympiads will regularly encounter rectangles in geometry and mensuration questions.
We explored rectangle—from its definition, formulas, properties, example problems, mistakes to avoid, and connections to other mathematical topics. Continue practicing with Vedantu for deeper understanding and exam success!
You may also like: Area of Rectangle | Perimeter of Rectangle | Properties of Rectangle | Difference Between Square and Rectangle
FAQs on Rectangle - Definition, Properties, and Formulas
1. What is the definition of a rectangle in geometry?
In geometry, a rectangle is defined as a quadrilateral (a four-sided polygon) that has four right angles (90°). A key characteristic is that its opposite sides are equal in length and parallel to each other.
2. What are the main properties of a rectangle?
The main properties of a rectangle are:
- It has four sides and four vertices.
- All four interior angles are exactly 90 degrees.
- Opposite sides are equal and parallel.
- The diagonals are equal in length.
- The diagonals bisect each other, meaning they intersect at their exact midpoint.
3. How do you calculate the area of a rectangle?
The area of a rectangle is calculated by multiplying its length (l) by its width (w). The formula is Area = l × w. The result is always expressed in square units, such as square centimetres (cm²) or square metres (m²).
4. What is the formula for finding the perimeter of a rectangle?
The perimeter of a rectangle is the total distance around its boundary. It is calculated using the formula Perimeter = 2(l + w), where 'l' represents the length and 'w' represents the width. This means you add the length and width first, then multiply the sum by two.
5. What is the difference between a rectangle and a square?
The main difference lies in the length of their sides. In a rectangle, only the opposite sides must be equal. In a square, all four sides must be equal in length. This means that every square is a rectangle, but not every rectangle is a square.
6. What are some real-world examples of rectangles?
Rectangles are one of the most common shapes found in the real world. Common examples include:
- The screens of televisions, laptops, and smartphones.
- Doors, windows, and bricks in construction.
- The pages of a notebook or a textbook.
- A standard swimming pool or a tennis court.
7. How can you find the length of a rectangle's diagonal?
The length of a rectangle's diagonal (d) can be found using the Pythagorean theorem. The formula is d = √(l² + w²). This means you square the length, square the width, add the two results, and then find the square root of the sum.
8. Why is a rectangle considered a special type of parallelogram?
A rectangle is a special parallelogram because it satisfies all the properties of a parallelogram (two pairs of parallel sides, opposite sides are equal) but includes one additional, specific condition: all its interior angles must be right angles (90°). A general parallelogram does not need to have right angles.
9. What happens to a rectangle's area if you double its length but keep the width the same?
If you double the length of a rectangle while keeping its width constant, the new area will also double. This is because the original area is l × w, and the new area becomes (2l) × w, which simplifies to 2 × (l × w).
10. Can a rectangle have diagonals that are not equal in length? Why or why not?
No, a rectangle cannot have diagonals of unequal length. It is a defining property of a rectangle that its diagonals must be equal. If a four-sided shape with four right angles were to have unequal diagonals, it would be a geometric impossibility. This property distinguishes it from other quadrilaterals like a non-rectangular parallelogram.
11. If a rectangle has an area of 36 sq. cm, what are its possible dimensions?
If the area is 36 sq. cm, the dimensions (length and width) must be two numbers that multiply to give 36. Assuming the dimensions are whole numbers, the possible pairs could be:
- 1 cm and 36 cm
- 2 cm and 18 cm
- 3 cm and 12 cm
- 4 cm and 9 cm
- 6 cm and 6 cm (which would make it a square)
