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Perimeter and Area

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concepts of perimeter and area for closed plane figures, focusing on rectangles, squares, and triangles. It exists in the syllabus to build foundational measurement skills needed for real-life applications like fencing, tiling, and carpeting. The one big idea a student must walk away with is that perimeter is the total distance around a shape's boundary, while area is the measure of the region enclosed by it, and these two quantities are independent—two shapes can have the same area but different perimeters, and vice versa.
Key Concepts (13)
Perimeter
The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once.
e.g. Walking around the boundary of a rectangular park once gives the perimeter of the park.
Perimeter of a polygon
The perimeter of a polygon is the sum of the lengths of all its sides.
e.g. A triangle with sides 4 cm, 5 cm, and 7 cm has a perimeter of 4 + 5 + 7 = 16 cm.
Perimeter of a rectangle
The perimeter of a rectangle is twice the sum of its length and breadth.
e.g. A rectangular tablecloth 3 m long and 2 m wide has a perimeter of 2 × (3 + 2) = 10 m.
Perimeter of a square
The perimeter of a square is quadruple the length of its side.
e.g. A square photo frame of side 1 m has a perimeter of 4 × 1 = 4 m.
Perimeter of a triangle
The perimeter of a triangle is the sum of the lengths of its three sides.
e.g. A triangle with sides 4 cm, 5 cm, and 7 cm has a perimeter of 16 cm.
Regular polygon
A closed figure that has all sides and all angles equal is called a regular polygon.
e.g. An equilateral triangle (all three sides equal, all three angles equal) is a regular polygon.
Perimeter of an equilateral triangle
The perimeter of an equilateral triangle is 3 times the length of a side.
e.g. An equilateral triangle with side 6 cm has perimeter 3 × 6 = 18 cm.
Area
The amount of region enclosed by a closed figure is called its area.
e.g. The floor of a room 5 m long and 4 m wide has an area of 20 square metres.
Area of a rectangle
The area of a rectangle is its length times its width.
e.g. A rectangular garden 25 m long and 12 m wide has an area of 25 × 12 = 300 sq m.
Area of a square
The area of a square is the length of any one of its sides multiplied by itself.
e.g. A square carpet of side 3 m has an area of 3 × 3 = 9 sq m.
Area of a triangle (half of rectangle)
The area of a triangle formed by drawing a diagonal of a rectangle is half the area of the rectangle.
e.g. If a rectangle has area 20 sq cm, the triangle formed by its diagonal has area 10 sq cm.
Estimating area using graph paper
To estimate the area of any simple closed shape, trace it on graph paper and count full squares as 1 sq unit, half squares as 1/2 sq unit, ignore less than half, and count more than half as 1 sq unit.
e.g. A leaf traced on graph paper covering 4 full squares, 2 half squares, and 3 more-than-half squares has an estimated area of 4 + 1 + 3 = 8 sq units.
Diagonal of a rectangle
A line segment joining two opposite vertices of a rectangle.
e.g. In rectangle ABCD, the diagonal AC divides it into two triangles of equal area.
Formulas & Equations (7)
Perimeter of a rectangle = 2 × (length + breadth)
Reads as: Perimeter of a rectangle equals two times the sum of length and breadth
Variables: length (l) in metres (m), breadth (b) in metres (m)
Use when: Finding the total length of boundary of a rectangular shape, e.g., fencing a rectangular park or putting lace around a tablecloth.
⚠ Students forget to multiply by 2 and write perimeter = length + breadth only.
Perimeter of a square = 4 × length of a side
Reads as: Perimeter of a square equals four times the length of a side
Variables: side (s) in metres (m)
Use when: Finding the boundary length of a square shape, e.g., tape around a square photo frame or distance around a square park.
⚠ Students multiply side by 2 instead of 4, confusing it with rectangle perimeter.
Perimeter of a triangle = sum of lengths of its three sides
Reads as: Perimeter of a triangle equals the sum of the lengths of its three sides
Variables: side1, side2, side3 in cm or m
Use when: Finding the total boundary length of any triangle, including equilateral triangles.
⚠ Students forget to add all three sides, especially when one side is missing and must be found.
Perimeter of an equilateral triangle = 3 × length of a side
Reads as: Perimeter of an equilateral triangle equals three times the length of a side
Variables: side (s) in cm or m
Use when: Finding the perimeter of a triangle where all sides are equal.
⚠ Students multiply by 4 (thinking of a square) instead of 3.
Area of a rectangle = length × breadth
Reads as: Area of a rectangle equals length times breadth
Variables: length (l) in metres (m), breadth (b) in metres (m); area in square metres (sq m)
Use when: Finding the region enclosed by a rectangle, e.g., area of a floor, garden, or land.
⚠ Students confuse area with perimeter and add length and breadth instead of multiplying.
Area of a square = side × side
Reads as: Area of a square equals side times side
Variables: side (s) in metres (m); area in square metres (sq m)
Use when: Finding the region enclosed by a square, e.g., area of a square carpet or flower bed.
⚠ Students multiply side by 4 (perimeter formula) instead of multiplying side by itself.
Area of a triangle = 1/2 × area of the rectangle it is part of
Reads as: Area of a triangle equals half the area of the rectangle it is part of
Variables: Area of rectangle in sq units; area of triangle in sq units
Use when: Finding the area of a triangle formed by drawing a diagonal of a rectangle or by splitting a rectangle.
⚠ Students forget to take half and write the full rectangle area as the triangle area.
How It Works
Finding perimeter of a rectangle
  1. Step 1: Identify the length and breadth of the rectangle.
  2. Step 2: Add the length and breadth.
  3. Step 3: Multiply the sum by 2.
  4. Step 4: Write the answer with the correct unit (e.g., cm, m).
💡 Students often forget to multiply by 2 and just add length and breadth. Always double-check.
Finding perimeter of a square
  1. Step 1: Identify the length of one side of the square.
  2. Step 2: Multiply that side length by 4.
  3. Step 3: Write the answer with the correct unit.
💡 Do not confuse with area — perimeter uses multiplication by 4, not squaring.
Finding perimeter of a triangle
  1. Step 1: Identify the lengths of all three sides.
  2. Step 2: Add the three side lengths together.
  3. Step 3: Write the answer with the correct unit.
💡 If one side is missing, subtract the sum of the two known sides from the total perimeter.
Finding area of a rectangle
  1. Step 1: Identify the length and breadth of the rectangle.
  2. Step 2: Multiply the length by the breadth.
  3. Step 3: Write the answer in square units (e.g., sq cm, sq m).
💡 Area is always in square units — never write just cm or m. Students often forget the 'square' part.
Finding area of a square
  1. Step 1: Identify the length of one side of the square.
  2. Step 2: Multiply the side by itself (side × side).
  3. Step 3: Write the answer in square units.
💡 Do not multiply by 4 — that gives perimeter, not area.
Estimating area using graph paper
  1. Step 1: Trace the shape onto transparent paper and place it on graph paper.
  2. Step 2: Count full squares inside the shape as 1 sq unit each.
  3. Step 3: Count half squares as 1/2 sq unit each.
  4. Step 4: Ignore portions less than half a square.
  5. Step 5: Count portions more than half a square as 1 sq unit each.
  6. Step 6: Add all counted values to get the estimated area.
💡 Students often count all partial squares as full squares, overestimating the area. Follow the rules strictly.
Finding area of a triangle by splitting a rectangle
  1. Step 1: Draw a diagonal of the rectangle to form two triangles.
  2. Step 2: Find the area of the rectangle using length × breadth.
  3. Step 3: Divide the rectangle area by 2 to get the area of one triangle.
  4. Step 4: Write the answer in square units.
💡 Both triangles formed by the diagonal have equal area. The area of each triangle is exactly half the rectangle area.
Exam Traps — Don't Make These Mistakes
⚠ Confusing perimeter with area
✗ For a rectangle of length 5 m and breadth 3 m, students write perimeter = 5 × 3 = 15 m.
→ They are multiplying length and breadth, which gives area, not perimeter. Perimeter requires addition and multiplication by 2.
✓ Perimeter = 2 × (5 + 3) = 2 × 8 = 16 m.
⚠ Forgetting to multiply by 2 in rectangle perimeter
✗ Perimeter = 5 + 3 = 8 m.
→ They only add length and breadth but forget to multiply the sum by 2. The perimeter must account for all four sides.
✓ Perimeter = 2 × (5 + 3) = 16 m.
⚠ Using perimeter formula for area of square
✗ Area of square with side 4 cm = 4 × 4 = 16 cm.
→ They wrote the correct number but the unit is wrong — area must be in square cm (sq cm), not cm.
✓ Area = 4 cm × 4 cm = 16 sq cm.
⚠ Counting all partial squares as full in graph paper area estimation
✗ Counted 5 full squares and 3 partial squares as 8 sq units.
→ Partial squares must be evaluated: less than half is ignored, exactly half is 1/2, more than half is 1. Counting all as 1 overestimates area.
✓ Count full squares = 5, half squares = 1 (count as 0.5), more than half = 2 (count as 2). Total = 5 + 0.5 + 2 = 7.5 sq units.
⚠ Assuming all triangles have area half of a rectangle
✗ Any triangle's area is half of the rectangle's area.
→ Only a triangle formed by drawing a diagonal of a rectangle (or one that exactly fills half the rectangle) has area half of that rectangle. Not all triangles fit this relationship.
✓ Only when a triangle is formed by the diagonal of a rectangle, its area is half the area of that rectangle.
⚠ Forgetting to convert units in perimeter and area problems
✗ Length = 1 m, breadth = 50 cm, perimeter = 2 × (1 + 50) = 102.
→ Units must be the same before adding or multiplying. 1 m = 100 cm, so correct sum is 100 + 50 = 150 cm.
✓ Convert 1 m to 100 cm, then perimeter = 2 × (100 + 50) = 2 × 150 = 300 cm.
Compare & Contrast
What they measure
Perimeter
Area
Perimeter measures the distance around the boundary of a closed figure (one-dimensional), while area measures the region enclosed by the figure (two-dimensional).
Formula
Perimeter of a rectangle
Area of a rectangle
Perimeter = 2 × (length + breadth) — you add then multiply by 2. Area = length × breadth — you multiply the two dimensions. They are never the same value unless the rectangle is a special case.
Side properties
Square
Rectangle
A square has all four sides equal, so its perimeter is 4 × side and area is side × side. A rectangle has opposite sides equal, so its perimeter is 2 × (length + breadth) and area is length × breadth.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (2-3 marks), 1 Long Answer (4-5 marks)
Most asked: Finding perimeter and area of rectangles and squares from given dimensions · Word problems involving fencing (perimeter) and tiling/carpeting (area) · Finding missing side when perimeter or area is given · Comparing perimeters and areas of different shapes
Don't skip: Word problems combining perimeter and area, e.g., 'A floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid. Find the uncarpeted area.'; Finding missing side when perimeter or area is given (e.g., 'Perimeter of rectangle = 14 cm, breadth = 2 cm, find length.'); Problems involving multiple rounds of fencing or multiple objects (e.g., 'A farmer fences a field with 3 rounds of rope.')
One-Line Revision (read aloud before the exam)
  1. Perimeter is the total distance around the boundary of a closed figure.
  2. Area is the amount of region enclosed by a closed figure.
  3. Perimeter of a rectangle = 2 × (length + breadth).
  4. Perimeter of a square = 4 × length of a side.
  5. Perimeter of a triangle = sum of lengths of its three sides.
  6. Perimeter of an equilateral triangle = 3 × length of a side.
  7. Area of a rectangle = length × breadth.
  8. Area of a square = side × side.
  9. Area of a triangle formed by a diagonal of a rectangle is half the area of the rectangle.
  10. Two figures can have the same area but different perimeters.
  11. Two figures can have the same perimeter but different areas.
  12. When estimating area on graph paper, count full squares as 1, half squares as 1/2, ignore less than half, and count more than half as 1.
  13. Always use the same unit before adding or multiplying in perimeter and area problems.
  14. Area is always expressed in square units (e.g., sq cm, sq m).

MCQs — 10 Questions with Answers

Q1. The perimeter of a shape is the total __________ around its boundary.

  • A. distance ✓
  • B. area
  • C. width
  • D. height

Answer: A — Perimeter is defined as the total distance measured along the outer boundary of a closed shape.

Q2. A square has a side of 7 cm. What is its perimeter?

  • A. 14 cm
  • B. 28 cm ✓
  • C. 49 cm
  • D. 7 cm

Answer: B — Perimeter of square = 4 × side = 4 × 7 = 28 cm.

Q3. A rectangle has length 10 m and breadth 6 m. Its perimeter is __________ m.

  • A. 16
  • B. 32 ✓
  • C. 60
  • D. 20

Answer: B — Perimeter = 2(l + b) = 2(10 + 6) = 2 × 16 = 32 m.

Q4. A triangle has sides of length 5 cm, 6 cm, and 9 cm. What is its perimeter?

  • A. 15 cm
  • B. 20 cm ✓
  • C. 30 cm
  • D. 24 cm

Answer: B — Perimeter of triangle = 5 + 6 + 9 = 20 cm.

Q5. A string of length 36 cm is bent to form a square. What is the side length of the square?

  • A. 6 cm
  • B. 9 cm ✓
  • C. 12 cm
  • D. 18 cm

Answer: B — If perimeter = 36 cm, then 4 × side = 36, so side = 36 ÷ 4 = 9 cm.

Q6. Which formula gives the perimeter of a rectangle with length l and breadth b?

  • A. l + b
  • B. l × b
  • C. 2(l + b) ✓
  • D. 2l + b

Answer: C — Perimeter of rectangle = 2 × (length + breadth) = 2(l + b) because opposite sides are equal.

Q7. Akshi runs around a rectangular park of length 80 m and breadth 50 m, completing 2 full rounds. What is the total distance she covers?

  • A. 130 m
  • B. 260 m
  • C. 520 m ✓
  • D. 4000 m

Answer: C — Perimeter = 2(80 + 50) = 260 m; distance in 2 rounds = 2 × 260 = 520 m.

Q8. A rectangular field has perimeter 100 m. If its length is 30 m, what is its breadth?

  • A. 10 m
  • B. 20 m ✓
  • C. 35 m
  • D. 70 m

Answer: B — Using P = 2(l + b): 100 = 2(30 + b), so 50 = 30 + b, thus b = 20 m.

Q9. In the figure shown, the outer rectangle has length 70 m and breadth 40 m. What is the distance Akshi covers in one complete round?

  • A. 110 m
  • B. 220 m ✓
  • C. 140 m
  • D. 2800 m

Answer: B — Perimeter of rectangle = 2(70 + 40) = 2 × 110 = 220 m, which is the distance in one round.

Q10. A regular hexagon (6-sided figure) has each side of length 5 cm. What is its perimeter?

  • A. 5 cm
  • B. 11 cm
  • C. 30 cm ✓
  • D. 36 cm

Answer: C — Perimeter of regular polygon = number of sides × length of one side = 6 × 5 = 30 cm.

Flashcards

What is perimeter?

Perimeter is the total distance around the outer boundary of any closed shape.

Formula for perimeter of a rectangle

Perimeter of rectangle = 2 × (length + breadth), or P = 2(l + b).

Formula for perimeter of a square

Perimeter of square = 4 × side, or P = 4s.

Formula for perimeter of a triangle

Perimeter of triangle = sum of all three sides, or P = a + b + c.

A rectangle is 8 m long and 5 m wide. What is its perimeter?

Perimeter = 2(8 + 5) = 2 × 13 = 26 metres.

A square has a side of 6 cm. What is its perimeter?

Perimeter = 4 × 6 = 24 cm.

A triangle has sides 5 cm, 7 cm, and 8 cm. What is its perimeter?

Perimeter = 5 + 7 + 8 = 20 cm.

If a wire of length 40 cm is bent into a square, what is the side length?

Side length = 40 ÷ 4 = 10 cm (because perimeter of square = 4 × side).

A rectangular park is 120 m long and 80 m wide. Usha runs around it 3 times. How far does she run?

Perimeter = 2(120 + 80) = 400 m; distance in 3 rounds = 3 × 400 = 1200 m.

Why do we use the formula 2(l + b) for rectangle instead of l + b + l + b?

Because opposite sides of a rectangle are equal in length, so we can write it as 2l + 2b = 2(l + b), which is shorter.

Important Board Questions

Define perimeter. [1 mark]

Perimeter is the total distance around the boundary of a closed shape; it is the sum of all side lengths.

A rectangular tablecloth is 4 m long and 3 m wide. Find the perimeter. Also, if lace costs ₹50 per metre, what is the cost of lacing the entire tablecloth? [2 marks]

Step 1: Calculate perimeter using P = 2(l + b) = 2(4 + 3). Step 2: Multiply perimeter by cost per metre to find total cost.

Ravi has a square garden with each side 12 m. He wants to fence it. The cost of fencing is ₹25 per metre. How much will he spend? Explain with calculation steps. [3 marks]

First find perimeter of square using P = 4 × side = 4 × 12 m. Then multiply by ₹25 per metre. Show all working steps clearly.

Two runners run on rectangular tracks. The outer track has length 70 m and breadth 40 m. The inner track has length 60 m and breadth 30 m. If the runner on the outer track completes 5 rounds and the runner on the inner track completes 8 rounds, who covers more distance and by how much? Show all your calculations. [5 marks]

Calculate perimeter of each rectangle using P = 2(l + b). Multiply each perimeter by number of rounds. Then find the difference to see who runs more.

True or False: The perimeter of a rectangle with length 8 cm and breadth 5 cm is 26 cm. Give reason for your answer. [2 marks]

Use formula P = 2(l + b) = 2(8 + 5) = 2 × 13 = 26 cm to check if statement is true or false; write clear reason.

Assertion-Reason Questions for Perimeter and Area

Q1. Assertion (A):

The perimeter of a rectangle with length 8 cm and breadth 5 cm is 26 cm.

Reason (R):

Perimeter of a rectangle = 2 × (length + breadth) = 2 × (8 + 5) = 2 × 13 = 26 cm.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the perimeter is 26 cm using the proper formula.

Q2. Assertion (A):

The perimeter of a square with side 7 m is 28 m.

Reason (R):

The perimeter of a square is equal to the area of the square.

Answer: C — A is true but R is false — The assertion is correct (4 × 7 = 28 m), but the reason is false because perimeter and area are different measurements. Perimeter is the distance around, while area is the space inside.

Q3. Assertion (A):

If Ravi runs 4 complete rounds around a rectangular park with length 60 m and breadth 40 m, he covers a total distance of 800 m.

Reason (R):

Because the perimeter of the park is 200 m and 4 rounds = 4 × 200 m = 800 m.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true. The perimeter = 2 × (60 + 40) = 200 m, and 4 rounds = 800 m, so the reason correctly explains the assertion.

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Case-Based Questions (CBQ) for Perimeter and Area

Case Study:

Neha's mother is making a decorative border around a square tablecloth using coloured ribbon. The tablecloth has a side length of 1.5 m. Neha wants to know how much ribbon her mother needs to buy. Her mother also wants to put the same ribbon around a rectangular bedsheet that is 2 m long and 1 m wide.

Q1. How much ribbon is needed for the square tablecloth?

  • A. 1.5 m
  • B. 6 m ✓
  • C. 2.25 m
  • D. 3 m

Answer: B

Q2. How much ribbon is needed for the rectangular bedsheet?

  • A. 2 m
  • B. 6 m ✓
  • C. 2 m²
  • D. 4 m

Answer: B

Q3. How much total ribbon does Neha's mother need to buy for both items?

  • A. 9 m
  • B. 12 m ✓
  • C. 10 m
  • D. 8 m

Answer: B

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