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Perimeter and Area

NCERT Class 5 · Mathematics Based on NCERT Class 5 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces the concepts of perimeter (the length of the border of a shape) and area (the region covered by a shape). It is important because these measurements are used in everyday life, such as decorating a quilt with lace or tiling a floor. The big idea is that perimeter is the total distance around a shape, while area is the number of unit squares that fit inside it, and for rectangles and squares, area can be found by multiplying length and breadth.
Key Concepts (10)
Perimeter
The length of the border of a shape is called its perimeter.
e.g. To put lace around a quilt, we measure the perimeter of the quilt to know how much lace is needed.
Area
The region covered by the triangles, squares or rectangles is called the area of the table.
e.g. Covering a table with square tiles — the number of tiles needed is the area of the table in square units.
Unit square
The area of a square with sides 1 unit is 1 unit square.
e.g. A square of side 1 cm has an area of 1 square centimetre (1 cm²).
Tiling
Filling a region with shapes that fit without gaps and overlaps.
e.g. Squares, rectangles, and triangles tile a surface; circles do not tile because they leave gaps.
Length
The longer side of a rectangle.
e.g. In a rectangle of 6 cm by 4 cm, the length is 6 cm.
Breadth
The shorter side of a rectangle.
e.g. In a rectangle of 6 cm by 4 cm, the breadth is 4 cm.
Square grid
A grid made of unit squares used to find area conveniently.
e.g. Placing a leaf on a square grid and counting the squares it covers gives its approximate area.
Rectangle
A shape with opposite sides equal and all angles right angles.
e.g. A notebook cover is usually a rectangle.
Square
A rectangle with all sides equal.
e.g. A square patchwork piece with side 5 cm.
Rows and columns
Arrangement of unit squares in a rectangle: number of rows × number of columns gives area.
e.g. A rectangle with 6 rows and 4 columns has area 6 × 4 = 24 square units.
Formulas & Equations (4)
Area of Rectangle = Length × Breadth
Reads as: Area of a rectangle equals length times breadth.
Variables: Length (L) in cm or m, Breadth (B) in cm or m, Area in square cm or square m.
Use when: When you need to find the area of a rectangular surface like a field, garden, or notebook cover.
⚠ Students often forget to write the unit as 'square cm' or 'square m' and just write 'cm' or 'm'.
Area of Square = Length × Length
Reads as: Area of a square equals side times side.
Variables: Length (side) in cm or m, Area in square cm or square m.
Use when: When finding the area of a square patch or square field.
⚠ Students sometimes multiply the side by 4 (which is perimeter) instead of by itself.
Perimeter of Rectangle = 2 × Length + 2 × Breadth
Reads as: Perimeter of a rectangle equals two times length plus two times breadth.
Variables: Length (L) and Breadth (B) in cm or m, Perimeter in cm or m.
Use when: When finding the total distance around a rectangular shape, like a garden or a room.
⚠ Students forget to add both lengths and both breadths, or they only add one length and one breadth.
Perimeter of Square = 4 × Length
Reads as: Perimeter of a square equals four times the side length.
Variables: Length (side) in cm or m, Perimeter in cm or m.
Use when: When finding the border length of a square shape.
⚠ Students confuse this with area and write side × side instead of 4 × side.
How It Works
Finding area by counting unit squares on a grid
  1. Place the shape on a square grid (each square is 1 unit × 1 unit).
  2. Count the number of full squares inside the shape.
  3. For partial squares, estimate or count half squares as 0.5.
  4. The total count is the area in unit squares.
💡 Always use the same unit for all measurements; partial squares must be estimated carefully.
Finding area of a rectangle by multiplication
  1. Measure the length and breadth of the rectangle in the same unit.
  2. Multiply the length by the breadth.
  3. Write the answer with the correct square unit (e.g., cm², m²).
💡 This method works only for rectangles and squares; for irregular shapes, use the grid method.
Finding perimeter by adding all sides
  1. Measure or note the length of each side of the shape.
  2. Add all the side lengths together.
  3. Write the answer with the same unit (e.g., cm, m).
💡 For rectangles, opposite sides are equal, so you can use the formula instead of adding all four sides individually.
Tiling a region without gaps and overlaps
  1. Choose a shape that tiles (squares, rectangles, triangles).
  2. Arrange the shapes edge-to-edge so no space is left uncovered.
  3. Count how many shapes are used — that number gives the area in that shape unit.
💡 Circles do not tile because they leave gaps; always use shapes that fit perfectly.
Exam Traps — Don't Make These Mistakes
⚠ Confusing perimeter with area
✗ The perimeter of a rectangle of length 6 cm and breadth 4 cm is 24 cm.
→ 24 cm is actually the area (6×4), not the perimeter. Perimeter is the sum of all sides: 6+4+6+4 = 20 cm.
✓ Perimeter = 2 × (6 + 4) = 20 cm.
⚠ Forgetting to write square units for area
✗ Area = 24 cm
→ Area is measured in square units, not linear units. Writing 'cm' implies length, not area.
✓ Area = 24 square cm (or 24 cm²).
⚠ Using wrong formula for square perimeter
✗ Perimeter of square with side 5 cm = 5 × 5 = 25 cm.
→ 5 × 5 gives area, not perimeter. Perimeter of a square is 4 × side.
✓ Perimeter = 4 × 5 = 20 cm.
⚠ Not counting partial squares correctly on a grid
✗ Area = number of full squares only, ignoring half squares.
→ Partial squares cover part of the region and must be counted to get the correct area.
✓ Count full squares as 1, half squares as 0.5, and estimate smaller parts.
⚠ Assuming all shapes with same area have same perimeter
✗ If two shapes have the same area, their perimeters are also equal.
→ Different shapes can have the same area but different perimeters (e.g., a long thin rectangle vs a square).
✓ Shapes with the same area can have different perimeters; always calculate separately.
Compare & Contrast
What they measure
Perimeter
Area
Perimeter measures the length of the boundary (distance around), while area measures the region inside (space covered).
Sides
Rectangle
Square
A rectangle has opposite sides equal; a square has all four sides equal.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Finding area and perimeter of rectangles and squares using formulas · Counting unit squares on a grid to find area · Word problems involving real-life situations (garden, field, quilt)
Don't skip: Word problem: Given area and one side of a rectangle, find the other side.; Compare areas of two shapes on a grid.; Find perimeter and area of a given figure with labelled sides.
One-Line Revision (read aloud before the exam)
  1. Perimeter is the total distance around a shape.
  2. Area is the region covered by a shape.
  3. Area of a rectangle = Length × Breadth.
  4. Area of a square = Side × Side.
  5. Perimeter of a rectangle = 2 × (Length + Breadth).
  6. Perimeter of a square = 4 × Side.
  7. A unit square has side 1 unit and area 1 unit square.
  8. Squares, rectangles, and triangles can tile a surface without gaps.
  9. Circles do not tile because they leave gaps.
  10. To find area on a grid, count the number of unit squares inside the shape.
  11. Different shapes can have the same area but different perimeters.
  12. Always write area in square units (e.g., cm², m²).
  13. Always write perimeter in linear units (e.g., cm, m).
  14. For a rectangle, opposite sides are equal in length.
  15. The product of rows and columns of unit squares gives the area of a rectangle.

MCQs — 10 Questions with Answers

Q1. What is the perimeter of a square with side 5 cm?

  • A. 20 cm ✓
  • B. 25 cm
  • C. 15 cm
  • D. 10 cm

Answer: A — Perimeter of square = 4 × side = 4 × 5 = 20 cm.

Q2. Which shape leaves gaps when tiling a table?

  • A. Rectangles
  • B. Circles ✓
  • C. Squares
  • D. Triangles

Answer: B — Circles have curved edges that leave gaps between them, so they cannot tile perfectly.

Q3. A rectangular garden has length 8 m and breadth 6 m. What is its area?

  • A. 14 square m
  • B. 28 square m
  • C. 48 square m ✓
  • D. 48 m

Answer: C — Area of rectangle = Length × Breadth = 8 × 6 = 48 square m.

Q4. Grandmother wants to put lace around the border of her quilt. Does she need to find the _____ or area?

  • A. perimeter ✓
  • B. area
  • C. length only
  • D. breadth only

Answer: A — Border means the edge around the quilt, which is the perimeter.

Q5. What is the area of a square with side 7 cm?

  • A. 28 square cm
  • B. 49 square cm ✓
  • C. 14 square cm
  • D. 35 square cm

Answer: B — Area of square = side × side = 7 × 7 = 49 square cm.

Q6. Two shapes have the same area. Will they always have the same perimeter?

  • A. Yes, always
  • B. No, they can have different perimeters ✓
  • C. Only if both are squares
  • D. Only if both are rectangles

Answer: B — Same area does not mean same perimeter; different shapes can have same area but different perimeters.

Q7. A rectangle has length 10 cm and breadth 4 cm. What is its perimeter?

  • A. 14 cm
  • B. 28 cm ✓
  • C. 40 cm
  • D. 20 cm

Answer: B — Perimeter = 2 × Length + 2 × Breadth = 2(10) + 2(4) = 20 + 8 = 28 cm.

Q8. To cover the floor of a classroom with square tiles, which do you need to find?

  • A. Perimeter of floor
  • B. Area of floor ✓
  • C. Length only
  • D. All sides separately

Answer: B — Covering the floor means filling the space inside, which requires area calculation.

Q9. In the figure below, the rectangle has length 12 cm and breadth 5 cm. What is its area?

  • A. 34 square cm
  • B. 60 square cm ✓
  • C. 17 square cm
  • D. 25 square cm

Answer: B — Area = Length × Breadth = 12 × 5 = 60 square cm.

Q10. Which measurement tells you how much lace is needed to decorate the border of a quilt cover?

  • A. Area
  • B. Perimeter ✓
  • C. Length only
  • D. Breadth only

Answer: B — The border is the outer edge, and the distance around it is the perimeter.

Flashcards

What is perimeter?

Perimeter is the total distance around the border of any shape.

What is area?

Area is the amount of space covered inside a shape.

Formula for perimeter of rectangle

Perimeter of rectangle = 2 × Length + 2 × Breadth or add all 4 sides.

Formula for area of rectangle

Area of rectangle = Length × Breadth.

Formula for perimeter of square

Perimeter of square = 4 × Length of one side.

Formula for area of square

Area of square = Length × Length.

What is 1 unit square?

1 unit square is a square with sides 1 unit long, used as a standard to measure area.

Which shapes tile without gaps?

Squares, rectangles, and triangles can tile together without gaps or overlaps.

Why use square grid to find area?

Square grids are the most convenient because squares have equal sides and fit together perfectly.

Can same area have different perimeter?

Yes, two shapes can have the same area but different perimeters.

Important Board Questions

What is perimeter? [1 mark]

It is the total distance around the outside border of any shape. Give the definition only.

A square has a side of 6 cm. Find its perimeter and area. [2 marks]

Use Perimeter of square = 4 × side and Area of square = side × side. Show both calculations.

Grandmother wants to put lace around her quilt border and also use patches to fill the quilt surface. Explain why she needs to find both perimeter and area, and give one example for each from your home. [3 marks]

Perimeter = distance around edge (for lace on border). Area = space inside (for patches covering). Use real examples like table, notebook, or floor.

A rectangular classroom has length 12 m and breadth 8 m. The teacher wants to put tiles on the floor and a border strip around the walls. Calculate (a) the area of the floor, (b) the perimeter of the floor, and (c) explain which measurement is needed for tiles and which for the border strip. [5 marks]

For area, use L × B. For perimeter, use 2L + 2B. Then explain: area for floor tiles (filling space), perimeter for border strip (around edge).

True or False: Two shapes can have the same area but different perimeters. Justify your answer with an example. [2 marks]

Think of different rectangles with same area but different length and breadth. Show one example with numbers and calculation.

Assertion-Reason Questions for Perimeter and Area

Q1. Assertion (A):

To cover a wall with tiles without any gaps, we should use circles because they are smooth.

Reason (R):

Circles leave gaps when placed next to each other, so they cannot tile a surface completely.

Answer: D — A is false but R is true — The assertion is false because circles do NOT tile without gaps. The reason is true, but since the assertion itself is wrong, the answer is D.

Q2. Assertion (A):

The perimeter of a square with sides of 4 cm is 16 cm.

Reason (R):

Perimeter of a square = 4 × side length = 4 × 4 = 16 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 16 cm. This is the correct formula and calculation.

Q3. Assertion (A):

Two rectangles with the same area must always have the same perimeter.

Reason (R):

A 2 cm × 8 cm rectangle and a 4 cm × 4 cm rectangle both have an area of 16 square cm, but their perimeters are different.

Answer: D — A is false but R is true — The assertion is false (two rectangles with the same area can have different perimeters), but the reason provided is true and proves the assertion wrong.

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Case-Based Questions (CBQ) for Perimeter and Area

Case Study:

Priya's mother is making a decorative frame for a rectangular picture. The picture is 40 cm long and 30 cm wide. She wants to put a golden border (lace) around the entire picture frame. She also needs to know how much glass she needs to cover the front of the picture, which is equal to the area of the picture.

Q1. How much lace does Priya's mother need to go around the entire border of the picture?

  • A. 70 cm
  • B. 140 cm ✓
  • C. 1200 cm
  • D. 120 cm

Answer: B

Q2. What is the area of the picture that needs to be covered with glass?

  • A. 70 square cm
  • B. 140 square cm
  • C. 1200 square cm ✓
  • D. 120 square cm

Answer: C

Q3. If the lace costs ₹5 per cm and the glass costs ₹10 per square cm, what is the total cost?

  • A. ₹700
  • B. ₹12700 ✓
  • C. ₹1700
  • D. ₹2400

Answer: B

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