5. Rectangle Formulas
A rectangle is a quadrilateral with four sides and four right angles. Its opposite sides are equal and parallel.
Let the length of the rectangle be l and its breadth be b.
The most important rectangle formulas are its area, perimeter and diagonal. Rectangle path questions are also commonly asked in competitive examinations.
What Is a Rectangle?
Opposite sides are equal and every angle is 90°.
1. Area of a Rectangle
The area of a rectangle is the product of its length and breadth.
Where:
- l = Length
- b = Breadth
The area is expressed in square units, such as cm², m² or km².
Example 1: Find the Area
A rectangular classroom has a length of 12 m and a breadth of 8 m. Find its area.
= 12 × 8
= 96 m²
Answer: 96 m²
Example 2: Find Length from Area
The area of a rectangle is 120 cm² and its breadth is 10 cm. Find its length.
From:
Therefore:
l = 120 ÷ 10
l = 12 cm
Answer: 12 cm
2. Perimeter of a Rectangle
A rectangle has two lengths and two breadths. Therefore, its perimeter is the total of all four sides.
The perimeter is expressed in ordinary length units such as cm, m or km.
Example
A rectangle has a length of 15 cm and breadth of 9 cm. Find its perimeter.
P = 2(15 + 9)
P = 2 × 24
P = 48 cm
Answer: 48 cm
If the question asks for fencing, boundary length or the distance around a rectangular field, use the perimeter formula.
3. Diagonal of a Rectangle
A diagonal joins two opposite corners of a rectangle.
Because the rectangle contains right angles, the diagonal can be found using the Pythagorean theorem.
Why?
The length, breadth and diagonal form a right-angled triangle. Therefore:
Taking the square root:
Example
A rectangle has a length of 12 cm and breadth of 5 cm. Find its diagonal.
d = √(12² + 5²)
d = √(144 + 25)
d = √169
d = 13 cm
Answer: 13 cm
4. Area of Path Inside a Rectangle
In some questions, a smaller rectangular region is removed from inside a larger rectangle, leaving a path of equal width around the inside.
x = width of the path
According to the source formula sheet, when the path has width x around the inside:
Explanation
The outer rectangle has dimensions l × b. After a path of width x is removed from all four sides, the dimensions of the inner rectangle become:
The path area is:
which simplifies to:
Example
A rectangular garden is 20 m long and 10 m broad. A path of width 1 m is made inside the garden along all four sides. Find the area of the path.
Here:
l = 20 m, b = 10 m, x = 1 m
= 2 × 1(20 + 10 - 2)
= 2 × 28
= 56 m²
Answer: 56 m²
5. Area of Path Outside a Rectangle
In this type of question, a path of width x is made outside the boundary of a rectangular region.
The shaded region represents the outside path.
The source formula gives:
Explanation
When the path is outside on all four sides, the new outer dimensions become:
Therefore:
which simplifies to:
Example
A rectangular field is 20 m long and 10 m broad. A path of width 1 m is constructed outside the field. Find the area of the path.
Here:
l = 20 m, b = 10 m, x = 1 m
= 2 × 1(20 + 10 + 2)
= 2 × 32
= 64 m²
Answer: 64 m²
6. Area of Path in the Middle of a Rectangle
Another common arrangement is a rectangular path or strip placed in the middle of a larger rectangle, leaving equal margins around it.
For the arrangement shown in the source, the path width is represented by x.
Example
A rectangular area has length 20 m and breadth 10 m. A middle path of width 2 m is formed according to the arrangement above. Find the area of the path.
Here:
l = 20 m, b = 10 m, x = 2 m
= 2(20 + 10 - 2)
= 2 × 28
= 56 m²
Answer: 56 m²
7. Rectangle Formula Chart
| Quantity | Formula |
|---|---|
| Area | l × b |
| Perimeter | 2(l + b) |
| Diagonal | √(l² + b²) |
| Path Inside | 2x(l + b - 2x) |
| Path Outside | 2x(l + b + 2x) |
| Path in the Middle | x(l + b - x) |
8. Useful Rearranged Formulas
From the area formula:
we can find the missing dimension:
From the perimeter formula:
we get:
Example
The perimeter of a rectangle is 50 cm and its length is 15 cm. Find its breadth.
50 = 2(15 + b)
25 = 15 + b
b = 10 cm
Answer: 10 cm
9. Common Mistakes in Rectangle Questions
Mistake 1: Confusing area and perimeter
Area uses:
Perimeter uses:
Mistake 2: Forgetting the square in the diagonal formula
Mistake 3: Using the wrong path formula
First identify whether the path is:
- Inside the rectangle
- Outside the rectangle
- In the middle of the rectangle
Mistake 4: Ignoring units
Area should be written in square units such as m², while perimeter and diagonal use ordinary length units such as m.
10. Real-Life Example
Consider a rectangular school playground measuring 50 m × 30 m.
To find the amount of ground covered:
To find the length of fencing required around it:
Thus, the same rectangle can require different formulas depending on what the question asks.
11. Practice Questions
Question 1: Find the area of a rectangle measuring 18 cm × 12 cm.
Question 2: Find the perimeter of a rectangle whose length is 25 m and breadth is 15 m.
Question 3: Find the diagonal of a rectangle measuring 9 cm × 12 cm.
Question 4: A rectangular garden is 30 m long and 20 m broad. Find the area of a 1 m wide path constructed inside it along all four sides.
Question 5: A rectangular field is 40 m long and 25 m broad. Find the area of a 2 m wide path constructed outside it.
Question 6: The perimeter of a rectangle is 70 cm and its length is 20 cm. Find its breadth.
12. Rectangle Formula Quick Revision
13. Mensuration Exam Tip
For quick revision, remember:
Area → Length × Breadth
Perimeter → 2(Length + Breadth)
Diagonal → √(Length² + Breadth²)
For path questions, carefully identify whether the path is inside, outside or in the middle before selecting the formula.
In Part 6 — Parallelogram Formulas, we will cover the area, perimeter and important diagonal relationships of a parallelogram, with diagrams, explanations, solved examples and exam tips.