Perimeter of Rectangle
Look around your classroom. The whiteboard on the wall, the door, your notebook, the top of your desk — all these are rectangular shapes. If you wanted to put a border or a frame around any of these, you would need to know the total length of the boundary. This total length of the boundary of a shape is called the perimeter.
The word perimeter comes from the Greek words "peri" (meaning around) and "metron" (meaning measure). So perimeter literally means "measure around."
Think about it this way: if an ant wants to walk all the way around your notebook, the distance it covers is the perimeter. If you want to stick decorative tape on the edges of a rectangular birthday card, the length of tape you need is the perimeter.
In this chapter, you will learn how to calculate the perimeter of a rectangle using its length and breadth. You will also learn how to find a missing side when the perimeter is known, and how to solve real-life word problems about fencing, framing, and bordering. This is one of the most useful formulas in everyday life. This topic is part of the Mensuration chapter in Grade 6 Maths (NCERT/CBSE).
What is Perimeter of Rectangle - Grade 6 Maths (Mensuration)?
Definition: The perimeter of a rectangle is the total distance around its boundary. It is the sum of all four sides.
A rectangle has:
- 4 sides
- Opposite sides are equal in length
- All 4 angles are right angles (90°)
- The longer side is usually called the length (l)
- The shorter side is usually called the breadth (b) (also called width)
Since a rectangle has two lengths and two breadths:
- Perimeter = length + breadth + length + breadth
- Perimeter = 2 × length + 2 × breadth
- Perimeter = 2 × (length + breadth)
Perimeter of Rectangle Formula
Formula:
Perimeter of Rectangle = 2 × (l + b)
Where:
- l = length of the rectangle
- b = breadth (width) of the rectangle
- The unit of perimeter is the same as the unit of length (cm, m, km, etc.)
Related formulas:
If perimeter and breadth are known: l = (P/2) − b
If perimeter and length are known: b = (P/2) − l
Derivation and Proof
A rectangle has 4 sides. The opposite sides are equal.
Label the sides:
- Top side = length (l)
- Bottom side = length (l)
- Left side = breadth (b)
- Right side = breadth (b)
Perimeter = sum of all sides
- Perimeter = l + b + l + b
- Perimeter = l + l + b + b
- Perimeter = 2l + 2b
- Perimeter = 2(l + b)
This is why we multiply the sum of length and breadth by 2 — because each measurement appears twice (once on each opposite side).
Think of it like walking around a rectangular park:
Start at one corner. Walk along the length. Turn and walk along the breadth. Turn and walk the length again. Turn and walk the breadth again. You are back where you started. The total distance you walked is the perimeter.
Types and Properties
Type 1: Find the perimeter given length and breadth
- Directly apply P = 2(l + b).
- Example: l = 12 cm, b = 8 cm → P = 2(12 + 8) = 2 × 20 = 40 cm
Type 2: Find a missing side when perimeter is known
- Use l = (P/2) − b or b = (P/2) − l.
- Example: P = 50 cm, l = 15 cm → b = (50/2) − 15 = 25 − 15 = 10 cm
Type 3: Word problems (fencing, framing, bordering)
- A farmer wants to fence a field, a student wants to put tape around a chart — find the total length of material needed.
Type 4: Cost-based problems
- Find the perimeter first, then multiply by the cost per unit length.
- Example: Cost of fencing = perimeter × rate per metre
Type 5: Unit conversion problems
- Length and breadth may be in different units. Convert to the same unit first, then find perimeter.
Solved Examples
Example 1: Example 1: Basic Perimeter Calculation
Problem: Find the perimeter of a rectangle with length 14 cm and breadth 9 cm.
Solution:
Given:
- Length (l) = 14 cm
- Breadth (b) = 9 cm
Using the formula:
- Perimeter = 2 × (l + b)
- Perimeter = 2 × (14 + 9)
- Perimeter = 2 × 23
- Perimeter = 46 cm
Answer: The perimeter is 46 cm.
Example 2: Example 2: Finding the Breadth
Problem: The perimeter of a rectangle is 64 cm. If the length is 20 cm, find the breadth.
Solution:
Given:
- Perimeter (P) = 64 cm
- Length (l) = 20 cm
Using the formula:
- P = 2(l + b)
- 64 = 2(20 + b)
- 64 ÷ 2 = 20 + b
- 32 = 20 + b
- b = 32 − 20
- b = 12 cm
Answer: The breadth is 12 cm.
Example 3: Example 3: Fencing a Garden
Problem: A rectangular garden is 25 m long and 18 m wide. How much fencing wire is needed to go around it completely?
Solution:
Given:
- Length = 25 m
- Breadth = 18 m
Fencing wire needed = Perimeter of the garden
- Perimeter = 2 × (25 + 18)
- Perimeter = 2 × 43
- Perimeter = 86 m
Answer: 86 metres of fencing wire is needed.
Example 4: Example 4: Cost of Fencing
Problem: A rectangular plot is 40 m long and 30 m wide. If fencing costs Rs. 25 per metre, find the total cost of fencing.
Solution:
Step 1: Find the perimeter.
- Perimeter = 2 × (40 + 30) = 2 × 70 = 140 m
Step 2: Find the total cost.
- Cost = Perimeter × Rate per metre
- Cost = 140 × 25
- Cost = Rs. 3,500
Answer: The total cost of fencing is Rs. 3,500.
Example 5: Example 5: Framing a Photograph
Problem: A photograph is 30 cm long and 20 cm wide. What length of frame material is needed to frame it?
Solution:
Given:
- Length = 30 cm
- Breadth = 20 cm
Frame material = Perimeter
- Perimeter = 2 × (30 + 20)
- Perimeter = 2 × 50
- Perimeter = 100 cm or 1 m
Answer: 100 cm (1 metre) of frame material is needed.
Example 6: Example 6: Finding the Length
Problem: The perimeter of a rectangular field is 180 m. If the breadth is 35 m, find the length.
Solution:
Given:
- Perimeter = 180 m
- Breadth = 35 m
Using the formula:
- P = 2(l + b)
- 180 = 2(l + 35)
- 90 = l + 35
- l = 90 − 35
- l = 55 m
Answer: The length is 55 m.
Example 7: Example 7: Ribbon Around a Gift Box Top
Problem: Meena wants to tie a ribbon around the top of a rectangular gift box that is 24 cm long and 16 cm wide. She needs an extra 20 cm for the bow. How long should the ribbon be?
Solution:
Step 1: Find the perimeter of the top.
- Perimeter = 2(24 + 16) = 2 × 40 = 80 cm
Step 2: Add extra for the bow.
- Total ribbon = 80 + 20 = 100 cm
Answer: Meena needs a ribbon of length 100 cm (1 m).
Example 8: Example 8: Unit Conversion Problem
Problem: A rectangular hall is 12 m long and 350 cm wide. Find its perimeter in metres.
Solution:
Step 1: Convert to the same unit.
- 350 cm = 350/100 m = 3.5 m
Step 2: Find perimeter.
- Perimeter = 2 × (12 + 3.5)
- Perimeter = 2 × 15.5
- Perimeter = 31 m
Answer: The perimeter is 31 m.
Example 9: Example 9: Multiple Rounds
Problem: A rectangular playground is 60 m long and 40 m wide. Rohit runs 3 rounds around it. How much distance does he cover?
Solution:
Step 1: Find perimeter (1 round).
- Perimeter = 2(60 + 40) = 2 × 100 = 200 m
Step 2: Distance for 3 rounds.
- Total distance = 3 × 200 = 600 m
Answer: Rohit covers 600 m.
Example 10: Example 10: Rectangle vs Square Perimeter
Problem: A rectangle has length 18 cm and breadth 12 cm. A square has side 15 cm. Which has a greater perimeter and by how much?
Solution:
Perimeter of rectangle:
- P = 2(18 + 12) = 2 × 30 = 60 cm
Perimeter of square:
- P = 4 × 15 = 60 cm
Answer: Both have the same perimeter of 60 cm.
Real-World Applications
The perimeter of a rectangle is used in many real-life situations:
- Fencing: To fence a rectangular garden, park, or plot of land, you need to know the perimeter to calculate how much wire or material to buy. A farmer fencing a rectangular field of 100 m by 50 m needs 2(100 + 50) = 300 m of fencing.
- Framing: When framing a picture, certificate, or mirror, you measure the perimeter to cut the right length of frame. A photo that is 20 cm by 15 cm needs 2(20 + 15) = 70 cm of frame material.
- Borders: Adding a decorative border to a rectangular chart paper or bulletin board — the perimeter tells you the length of border tape needed.
- Running tracks: The distance around a rectangular playground or sports court is its perimeter. One full round = one perimeter. If a student runs 5 rounds of a 80 m by 40 m playground, the total distance is 5 × 2(80 + 40) = 5 × 240 = 1200 m.
- Construction: Builders calculate the perimeter of rooms to estimate the length of skirting, wall trim, or baseboard moulding needed around a room.
- Ribbon and lace: To tie ribbon around a gift box or attach lace to a tablecloth, the perimeter tells you how much you need.
- Garden path: If you want to lay tiles along the boundary of a rectangular garden, you need to know the perimeter to calculate how many tiles to buy.
- Rope for a boundary: In school sports days, the rectangular playing area is often marked with rope. The length of rope equals the perimeter of the rectangle.
Key Points to Remember
- The perimeter of a shape is the total distance around its boundary.
- A rectangle has two lengths and two breadths, with opposite sides equal.
- Perimeter of a rectangle = 2 × (length + breadth) = 2(l + b).
- The unit of perimeter is the same as the unit of the sides (cm, m, km, etc.).
- To find a missing side: l = (P/2) − b or b = (P/2) − l.
- In word problems, the perimeter tells you the total length of material needed to go around the rectangle.
- Always check that length and breadth are in the same unit before applying the formula.
- Perimeter measures length (1-dimensional), not area (2-dimensional).
- For cost problems: Total cost = Perimeter × Cost per unit length.
- One full round around a rectangle = its perimeter. For n rounds, distance = n × perimeter.
Practice Problems
- Find the perimeter of a rectangle with length 22 cm and breadth 13 cm.
- The perimeter of a rectangle is 96 m. If its length is 28 m, find the breadth.
- A rectangular park is 80 m long and 55 m wide. Find the cost of fencing it at Rs. 30 per metre.
- How many metres of ribbon are needed to go around a rectangular box top that is 35 cm by 20 cm? Give your answer in metres.
- A farmer walks around his rectangular field twice. The field is 100 m long and 60 m wide. What total distance does he walk?
- The perimeter of a rectangular swimming pool is 120 m. If its breadth is 20 m, find its length.
- A rectangular room is 5 m long and 4 m wide. Find the cost of fixing a border strip around the room at Rs. 50 per metre.
- Which has a greater perimeter: a rectangle with l = 25 cm, b = 10 cm, or a rectangle with l = 20 cm, b = 16 cm?
Frequently Asked Questions
Q1. What is the perimeter of a rectangle?
The perimeter of a rectangle is the total length of all four sides. Since opposite sides of a rectangle are equal, the formula is Perimeter = 2 × (length + breadth).
Q2. What is the formula for the perimeter of a rectangle?
Perimeter = 2(l + b), where l is the length and b is the breadth. You add the length and breadth, then multiply by 2.
Q3. What is the difference between perimeter and area?
Perimeter is the total length around the boundary of a shape (measured in cm, m, etc.). Area is the space inside the shape (measured in sq. cm, sq. m, etc.). Perimeter is 1-dimensional; area is 2-dimensional.
Q4. Can two different rectangles have the same perimeter?
Yes. For example, a rectangle with l = 10 cm, b = 5 cm has perimeter 30 cm. A rectangle with l = 8 cm, b = 7 cm also has perimeter 30 cm. Many different rectangles can share the same perimeter.
Q5. How do you find the length if perimeter and breadth are given?
Use the formula: length = (Perimeter / 2) − breadth. First divide the perimeter by 2 to get the sum of one length and one breadth, then subtract the breadth to get the length.
Q6. What units is perimeter measured in?
Perimeter is measured in units of length: centimetres (cm), metres (m), kilometres (km), etc. It is NOT measured in square units (that is for area).
Q7. What happens if length and breadth are in different units?
You must convert them to the same unit before calculating. For example, if length = 2 m and breadth = 50 cm, convert breadth to metres: 50 cm = 0.5 m. Then Perimeter = 2(2 + 0.5) = 5 m.
Q8. Is the perimeter of a square a special case of rectangle perimeter?
Yes. A square is a rectangle where length equals breadth (l = b = s). So Perimeter = 2(s + s) = 2 × 2s = 4s. This gives the familiar formula: Perimeter of square = 4 × side.
