Area and Its Boundary — Class 5 Mathematics (CBSE)

Based on the NCERT Math Magic Grade 5 textbook. Find area and perimeter of shapes, then solve the practice set without looking at the answers.

1. Why this chapter matters

How much land does a farmer need to grow crops? How much paint is needed to cover a wall? How long a fence goes around a garden? These are questions of area and perimeter. This chapter builds on earlier concepts (counting squares) and introduces formulas for the area and perimeter of squares and rectangles. Students also learn to estimate the area of irregular shapes using a grid. This is practical mathematics used in construction, farming, interior design, and planning.

2. Perimeter review

Perimeter is the total distance around a shape. It is found by adding the lengths of all sides.

Formula for perimeter

• Square: Perimeter = 4 x side
• Rectangle: Perimeter = 2 x (length + breadth)
• Triangle: Perimeter = side 1 + side 2 + side 3

3. Area of squares and rectangles

Area is the amount of surface a shape covers.

Formulas

• Square: Area = side x side
• Rectangle: Area = length x breadth
• Units: square centimetres (sq cm or cm^2), square metres (sq m or m^2)

Finding length or breadth given area

If area = 48 sq cm and length = 8 cm, then breadth = 48 / 8 = 6 cm. If area = 64 sq cm and it is a square, then side = square root of 64 = 8 cm.

4. Relationship between perimeter and area

Same perimeter does not mean same area. Same area does not mean same perimeter.

Same perimeter, different areas

For the same perimeter, a square gives the maximum area.

Same area, different perimeters

For the same area, the shape closest to a square gives the smallest perimeter.

5. Area of irregular shapes using grid

Not all shapes are squares or rectangles. To find the area of an irregular shape:

1. Place the shape on a grid (1 cm x 1 cm).
2. Count all full squares inside the boundary: 1 each.
3. Count half squares: two halves = 1 full square.
4. Count squares that are more than half-filled: count as 1.
5. Count squares that are less than half-filled: ignore.
6. Add all counts to get the approximate area.

Example: An irregular leaf on a grid:

• Full squares: 12
• Half squares: 6 (= 3 full squares)
• More than half: 4
• Less than half: 5 (ignore)
• Approximate area = 12 + 3 + 4 = 19 sq cm

6. Word problems

Problem 1: A rectangular garden is 12 m long and 8 m wide. Find its area and the length of fencing needed. Area = 12 x 8 = 96 sq m. Perimeter (fencing) = 2 x (12 + 8) = 40 m.

Problem 2: A square tile has side 10 cm. How many tiles are needed to cover a floor of length 200 cm and breadth 150 cm? Area of one tile = 10 x 10 = 100 sq cm. Area of floor = 200 x 150 = 30,000 sq cm. Number of tiles = 30,000 / 100 = 300 tiles.

Problem 3: A farmer has a rectangular field of 50 m by 30 m. He wants to increase the length by 5 m. How much does the area increase? Original area = 50 x 30 = 1500 sq m. New area = 55 x 30 = 1650 sq m. Increase = 1650 — 1500 = 150 sq m.

7. Activity corner

Activity 1: Find the area of your maths textbook cover by measuring its length and breadth in centimetres. Also find its perimeter.

Activity 2: On a 1 cm grid paper, draw 3 different rectangles each with an area of 24 sq cm. Calculate the perimeter of each.

Activity 3: Trace your hand on a 1 cm grid paper. Count the squares to find the approximate area of your hand.

8. Common mistakes

• Mistake: Confusing area formula with perimeter formula Fix: Area multiplies (length x breadth). Perimeter adds (2 x (L + B)).
• Mistake: Using different units for length and breadth Fix: Always convert to the same unit before calculating area or perimeter.
• Mistake: Forgetting to write square units for area Fix: Area is in square units (sq cm, sq m). Perimeter is in units (cm, m). Write both correctly.
• Mistake: Counting only two sides for perimeter Fix: Remember to add all four sides. For a rectangle, use 2 x (L + B).

9. Key facts

• Square area = side x side. Square perimeter = 4 x side.
• Rectangle area = length x breadth. Rectangle perimeter = 2 x (L + B).
• Same perimeter can give different areas. Same area can give different perimeters.
• For irregular shapes, use a grid to estimate area.
• Always use the same unit for all measurements.
• Area is in square units; perimeter is in linear units.

10. Self-test

1. Find the area of a square with side 9 cm.
2. A rectangle has length 14 cm and breadth 6 cm. Find its perimeter and area.
3. A rectangular plot is 20 m by 15 m. Find the cost of fencing at Rs 5 per metre.
4. Which has a larger area: a square of side 8 m or a rectangle of 10 m by 6 m?
5. An irregular shape on a grid has 15 full squares, 8 half squares, and 3 more-than-half squares. Find its approximate area.

11. Answer key

1. Find the area of a square with side 9 cm. Answer: Area = 9 x 9 = 81 sq cm.

2. A rectangle has length 14 cm and breadth 6 cm. Find its perimeter and area. Answer: Perimeter = 2 x (14 + 6) = 40 cm. Area = 14 x 6 = 84 sq cm.

3. A rectangular plot is 20 m by 15 m. Find the cost of fencing at Rs 5 per metre. Answer: Perimeter = 2 x (20 + 15) = 70 m. Cost = 70 x 5 = Rs 350.

4. Which has a larger area: a square of side 8 m or a rectangle of 10 m by 6 m? Answer: Square area = 64 sq m. Rectangle area = 60 sq m. The square has a larger area.

5. An irregular shape on a grid has 15 full squares, 8 half squares, and 3 more-than-half squares. Find its approximate area. Answer: Area ≈ 15 + (8/2) + 3 = 15 + 4 + 3 = 22 sq cm.

12. Quick revision

• Area of square = side x side. Perimeter of square = 4 x side.
• Area of rectangle = L x B. Perimeter of rectangle = 2 x (L + B).
• Use grid to find area of irregular shapes.
• Same perimeter does not mean same area.
• Check units carefully before calculating.
• Practise with real objects — measure books, tables, floors.