The Area Of A Rectangle Formula is one of the most fundamental expressions in geometry, stated as Area = Length × Breadth, and it appears in NCERT Mathematics from Class 4 right through to Class 10 and beyond. This formula is essential for CBSE board exams, and its applications extend into mensuration problems tested in JEE Main and NEET. In this article, we cover the formula, its derivation, a complete formula sheet, three progressively difficult solved examples, CBSE exam tips for 2025-26, common mistakes, and JEE/NEET applications.
Key Area Of A Rectangle Formulas at a Glance
Quick reference for the most important rectangle formulas you need to know.
- Area of a Rectangle: \( A = l \times b \)
- Perimeter of a Rectangle: \( P = 2(l + b) \)
- Diagonal of a Rectangle: \( d = \sqrt{l^2 + b^2} \)
- Area using diagonal: \( A = \frac{d^2 \sin\theta}{2} \) (where \( \theta \) is the angle between diagonals)
- Length from Area: \( l = \frac{A}{b} \)
- Breadth from Area: \( b = \frac{A}{l} \)
- Area of a Square (special rectangle): \( A = a^2 \)
What is the Area Of A Rectangle Formula?
The Area Of A Rectangle Formula gives the total two-dimensional space enclosed within the four sides of a rectangle. A rectangle is a quadrilateral with four right angles (each measuring 90°) and two pairs of equal, parallel sides. The longer side is called the length and the shorter side is called the breadth (or width).
In NCERT Mathematics, this formula is first introduced in Class 4 (Chapter: Shapes and Space) and is revisited with greater depth in Class 6 (Chapter 10: Mensuration) and Class 7 (Chapter 11: Perimeter and Area). It forms the foundation for more complex area calculations in Class 9 and Class 10.
The area of a rectangle is always expressed in square units. If the length and breadth are measured in centimetres, the area is in cm². If measured in metres, the area is in m². This concept is referenced in the NCERT Class 6 Maths textbook under Chapter 10 and is a compulsory topic for CBSE board exams.
Understanding this formula thoroughly helps students tackle mensuration, coordinate geometry, and even integration problems in higher classes.
Area Of A Rectangle Formula — Expression and Variables
The standard expression for the area of a rectangle is:
\[ A = l \times b \]
In plain text: Area = Length × Breadth
| Symbol | Quantity | SI Unit |
|---|---|---|
| \( A \) | Area of the rectangle | Square metre (m²) or cm² |
| \( l \) | Length (longer side) | Metre (m) or centimetre (cm) |
| \( b \) | Breadth or Width (shorter side) | Metre (m) or centimetre (cm) |
Derivation of the Area Of A Rectangle Formula
Consider a rectangle of length \( l \) and breadth \( b \). We can fill this rectangle with unit squares, each of area 1 square unit.
Step 1: Place unit squares along the length. You can fit exactly \( l \) unit squares in one row.
Step 2: Stack such rows along the breadth. You can fit exactly \( b \) rows.
Step 3: Total unit squares = \( l \times b \).
Step 4: Since each unit square has an area of 1 square unit, the total area is \( A = l \times b \).
This counting argument, rooted in the concept of measurement, confirms the formula. For non-integer dimensions, the same result holds by extension of the real number system.
Area of a Rectangle by Diagonal
Sometimes, the length and breadth are not directly given. Instead, the diagonal and one side (or the angle between the diagonals) may be provided. The diagonal \( d \) of a rectangle is related to its sides by the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
If you know the diagonal \( d \) and one side, say the length \( l \), you can find the breadth:
\[ b = \sqrt{d^2 – l^2} \]
Then the area becomes:
\[ A = l \times \sqrt{d^2 – l^2} \]
Alternatively, if the two diagonals of a rectangle intersect at an angle \( \theta \), the area can be expressed as:
\[ A = \frac{d^2 \sin\theta}{2} \]
This form is particularly useful in JEE problems where the diagonal and intersection angle are given instead of the sides directly.
Complete Mensuration and Algebra Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Area of a Rectangle | \( A = l \times b \) | l = length, b = breadth | m² or cm² | Class 6, Ch 10 |
| Perimeter of a Rectangle | \( P = 2(l + b) \) | l = length, b = breadth | m or cm | Class 6, Ch 10 |
| Diagonal of a Rectangle | \( d = \sqrt{l^2 + b^2} \) | l = length, b = breadth | m or cm | Class 9, Ch 7 |
| Area of a Square | \( A = a^2 \) | a = side length | m² or cm² | Class 6, Ch 10 |
| Perimeter of a Square | \( P = 4a \) | a = side length | m or cm | Class 6, Ch 10 |
| Area of a Triangle | \( A = \frac{1}{2} \times b \times h \) | b = base, h = height | m² or cm² | Class 7, Ch 11 |
| Area of a Circle | \( A = \pi r^2 \) | r = radius | m² or cm² | Class 7, Ch 11 |
| Area of a Parallelogram | \( A = b \times h \) | b = base, h = perpendicular height | m² or cm² | Class 9, Ch 9 |
| Area of a Trapezium | \( A = \frac{1}{2}(a + b) \times h \) | a, b = parallel sides, h = height | m² or cm² | Class 8, Ch 11 |
| Area of a Rhombus | \( A = \frac{1}{2} d_1 d_2 \) | d₁, d₂ = diagonals | m² or cm² | Class 8, Ch 11 |
Area Of A Rectangle Formula — Solved Examples
Example 1 (Class 6-8 Level)
Problem: A rectangular garden has a length of 15 m and a breadth of 8 m. Find the area of the garden.
Given: Length \( l = 15 \) m, Breadth \( b = 8 \) m
Step 1: Write the formula: \( A = l \times b \)
Step 2: Substitute the values: \( A = 15 \times 8 \)
Step 3: Calculate: \( A = 120 \) m²
Answer
The area of the rectangular garden is 120 m².
Example 2 (Class 9-10 Level)
Problem: The area of a rectangle is 360 cm². Its length is three times its breadth. Find the length, breadth, and the perimeter of the rectangle.
Given: \( A = 360 \) cm², \( l = 3b \)
Step 1: Write the area formula: \( A = l \times b \)
Step 2: Substitute \( l = 3b \): \( 360 = 3b \times b = 3b^2 \)
Step 3: Solve for \( b \): \( b^2 = \frac{360}{3} = 120 \), so \( b = \sqrt{120} = 2\sqrt{30} \approx 10.95 \) cm
Step 4: Find \( l \): \( l = 3b = 3 \times 2\sqrt{30} = 6\sqrt{30} \approx 32.86 \) cm
Step 5: Find the perimeter: \( P = 2(l + b) = 2(6\sqrt{30} + 2\sqrt{30}) = 2 \times 8\sqrt{30} = 16\sqrt{30} \approx 87.6 \) cm
Answer
Breadth \( = 2\sqrt{30} \approx 10.95 \) cm, Length \( = 6\sqrt{30} \approx 32.86 \) cm, Perimeter \( = 16\sqrt{30} \approx 87.6 \) cm.
Example 3 (JEE/NEET Level)
Problem: A rectangle has a diagonal of length 13 cm. The perimeter of the rectangle is 34 cm. Find the area of the rectangle.
Given: Diagonal \( d = 13 \) cm, Perimeter \( P = 34 \) cm
Step 1: From the perimeter formula: \( 2(l + b) = 34 \), so \( l + b = 17 \) … (i)
Step 2: From the diagonal formula: \( l^2 + b^2 = d^2 = 13^2 = 169 \) … (ii)
Step 3: Use the algebraic identity: \( (l + b)^2 = l^2 + 2lb + b^2 \)
Step 4: Substitute from (i) and (ii): \( 17^2 = 169 + 2lb \), so \( 289 = 169 + 2lb \)
Step 5: Solve for \( lb \): \( 2lb = 289 – 169 = 120 \), so \( lb = 60 \)
Step 6: Since \( A = l \times b = lb = 60 \) cm²
Answer
The area of the rectangle is 60 cm². (Verification: \( l = 12 \) cm, \( b = 5 \) cm satisfies all conditions.)
CBSE Exam Tips 2025-26
- Always state the formula first. In CBSE board exams 2025-26, writing the formula before substituting values earns you method marks even if the final answer is wrong.
- Write units at every step. We recommend writing cm² or m² alongside each intermediate calculation. Missing units in the final answer costs half a mark in many CBSE marking schemes.
- Convert units before calculating. If length is given in metres and breadth in centimetres, convert both to the same unit first. This is a very common error in Class 6-8 exams.
- For word problems, draw a labelled diagram. Our experts suggest drawing a rectangle and labelling \( l \) and \( b \) clearly. This helps you avoid confusion between length and breadth.
- Memorise the diagonal formula. The CBSE Class 9-10 syllabus frequently combines the area formula with the Pythagorean theorem. Knowing \( d = \sqrt{l^2 + b^2} \) saves time in exams.
- Check reasonableness of your answer. Area must always be a positive value. If you get a negative or zero result, recheck your substitution immediately.
Common Mistakes to Avoid
- Confusing Area with Perimeter: Many students use \( 2(l + b) \) when asked for area. Remember — area is \( l \times b \) and perimeter is \( 2(l + b) \). These are two completely different quantities.
- Forgetting to square the units: If length is in cm, area is in cm², not cm. Always square the unit when writing the final answer for area.
- Using the diagonal as a side: Some students mistakenly use the diagonal value as the length or breadth. The diagonal is the hypotenuse of the right triangle formed by \( l \) and \( b \), not a side of the rectangle.
- Not converting mixed units: When one dimension is in metres and another is in centimetres, students often multiply without converting. Always ensure both dimensions are in the same unit before applying \( A = l \times b \).
- Applying the square formula to a rectangle: A square is a special rectangle where \( l = b = a \), so \( A = a^2 \). This formula does NOT apply to a general rectangle where \( l \neq b \).
JEE/NEET Application of the Area Of A Rectangle Formula
In our experience, JEE aspirants encounter the Area Of A Rectangle Formula most frequently in coordinate geometry and integration problems. NEET students encounter it in Physics problems involving pressure on surfaces and in Biology diagrams involving cross-sectional areas.
Application Pattern 1: Coordinate Geometry (JEE Main)
JEE Main regularly asks students to find the area of a rectangle formed by four given coordinate points. If the vertices are \( (x_1, y_1) \), \( (x_2, y_1) \), \( (x_2, y_2) \), and \( (x_1, y_2) \), then the length is \( |x_2 – x_1| \) and the breadth is \( |y_2 – y_1| \). The area becomes:
\[ A = |x_2 – x_1| \times |y_2 – y_1| \]
Application Pattern 2: Optimisation Problems (JEE Advanced)
JEE Advanced often presents problems where a rectangle is inscribed in a circle or triangle, and you must maximise the area. For a rectangle inscribed in a circle of radius \( r \), the sides satisfy \( l^2 + b^2 = (2r)^2 \). Using calculus or the AM-GM inequality, the maximum area is \( A_{\max} = 2r^2 \), achieved when the rectangle is a square (\( l = b = r\sqrt{2} \)).
Application Pattern 3: Physics — Pressure and Force (NEET/JEE)
In Physics, pressure is defined as force per unit area: \( P = F/A \). When a force acts on a rectangular surface, the area \( A = l \times b \) is used directly. NEET problems on fluid pressure and stress in materials frequently require computing the rectangular cross-sectional area before applying the pressure formula.
In our experience, JEE aspirants who are comfortable with the rectangle area formula and its extensions (diagonal, inscribed shapes, coordinate form) consistently score higher in the mensuration and coordinate geometry sections.
FAQs on Area Of A Rectangle Formula
We hope this comprehensive guide to the Area Of A Rectangle Formula has helped you build a strong foundation. For more geometry and algebra formulas, explore our Algebra Formulas hub. You may also find our articles on the Pythagorean Theorem Formula and the Mean Median Mode Formula useful for your CBSE and competitive exam preparation. For official NCERT textbook content, you can refer to the NCERT official website.