How Many Squares — Class 5 Mathematics (CBSE)
Based on the NCERT Math Magic Grade 5 textbook. Learn to measure area by counting squares, then solve the practice set without looking at the answers.
1. Why this chapter matters
Area and perimeter are everyday concepts. How much space does your desk cover? How much lace is needed around a handkerchief? This chapter answers these questions by teaching students to find area by counting unit squares and perimeter by adding side lengths. Students also explore how different shapes can have the same perimeter but different areas, and vice versa. Tangrams add a fun, creative element to understanding area.
2. What is area?
Area is the amount of surface a shape covers. In Class 5, we measure area by counting the number of unit squares that fit inside a shape.
- Each small square is called a unit square (usually 1 cm x 1 cm).
- The area is the total number of unit squares that cover the shape completely.
- Area is expressed in square units (sq cm, sq m).
Counting squares to find area
Place a shape on a grid of 1 cm x 1 cm squares. Count the number of squares the shape covers.
- Count full squares as 1 each.
- Count half squares as 1/2 each.
- For irregular shapes, combine two halves to make one full square.
Example: A rectangle on a grid that covers 6 full squares has an area of 6 square centimetres (6 sq cm).
3. What is perimeter?
Perimeter is the total distance around the outside of a shape. It is found by adding the lengths of all the sides.
- Perimeter = Sum of all side lengths
- Perimeter is expressed in units of length (cm, m)
Example: A rectangle with length 5 cm and breadth 3 cm has:
- Perimeter = 5 + 3 + 5 + 3 = 16 cm
- Area (by counting squares on a grid) = 5 x 3 = 15 sq cm
Comparison table
| Shape | Length | Breadth | Perimeter | Area |
|---|---|---|---|---|
| Rectangle A | 6 cm | 2 cm | 16 cm | 12 sq cm |
| Rectangle B | 4 cm | 4 cm | 16 cm | 16 sq cm |
| Rectangle C | 5 cm | 3 cm | 16 cm | 15 sq cm |
Notice: All three rectangles have the same perimeter (16 cm) but different areas. A square (special rectangle) gives the maximum area for a given perimeter.
4. Different rectangles with same area, different perimeter
Now let us try the reverse.
| Shape | Length | Breadth | Area | Perimeter |
|---|---|---|---|---|
| Rectangle P | 12 cm | 1 cm | 12 sq cm | 26 cm |
| Rectangle Q | 6 cm | 2 cm | 12 sq cm | 16 cm |
| Rectangle R | 4 cm | 3 cm | 12 sq cm | 14 cm |
Notice: All three rectangles have the same area (12 sq cm) but different perimeters. The shape closest to a square has the smallest perimeter.
5. Tangrams
A tangram is a Chinese puzzle made by cutting a square into seven pieces:
- 2 large right-angled triangles
- 1 medium right-angled triangle
- 2 small right-angled triangles
- 1 square
- 1 parallelogram
Tangram rules
- Use all seven pieces to form a shape.
- Pieces must not overlap.
- Pieces must touch each other.
Learning from tangrams
Tangrams help develop spatial reasoning. By rearranging the same seven pieces, students discover that different shapes can have the same total area. This reinforces the idea that area is conserved even when shape changes.
Activity: Use a tangram set to create a cat, a house, a boat, or a bird. Count the squares covered by each piece to verify the total area.
6. Area of irregular shapes
Not all shapes are perfect rectangles. To find the area of an irregular shape on a grid:
- Count all full squares inside the shape.
- Count half squares — two halves make one full square.
- Count squares that are more than half as 1, ignore squares that are less than half.
- Add all counts to get the approximate area.
Example: A leaf drawn on a grid covers 8 full squares, 4 half squares, and 3 squares that are more than half. The area is approximately 8 + (4/2) + 3 = 8 + 2 + 3 = 13 sq cm.
7. Activity corner
Activity 1: Draw any five rectangles on a 1 cm grid paper. Count the squares to find the area of each. Then calculate the perimeter by adding the sides.
Activity 2: Using 12 unit squares, arrange them in different rectangles. How many different rectangles can you make? What are their perimeters?
Activity 3: Make your own tangram set by cutting a square piece of paper into seven pieces as described above. Create three different animal shapes using all seven pieces.
8. Common mistakes
- Mistake: Counting the squares on the border instead of inside the shape Fix: Shade the area inside the shape first, then count only the shaded squares.
- Mistake: Forgetting to add all four sides for perimeter Fix: Write the length of each side in order and add. Check that opposite sides are equal in a rectangle.
- Mistake: Confusing area and perimeter units Fix: Area is in square units (sq cm), perimeter is in units (cm). Always write the correct unit.
9. Key facts
- Area = number of unit squares covering a shape.
- Perimeter = sum of all side lengths.
- Same perimeter does not mean same area.
- Same area does not mean same perimeter.
- A square gives the largest area for a given perimeter.
- Tangrams use seven pieces to make infinite shapes — total area stays the same.
- 1 square centimetre = area of a square of side 1 cm.
10. Self-test
- What is the area of a rectangle that covers 8 full squares on a 1 cm grid?
- Find the perimeter of a square with side 5 cm.
- Two rectangles have the same perimeter of 20 cm. Give one possible pair of lengths and breadths.
- How many tangram pieces are there? Name them.
- A leaf drawn on a grid covers 10 full squares, 6 half squares, and 2 big squares (more than half). Find the approximate area.
11. Answer key
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What is the area of a rectangle that covers 8 full squares on a 1 cm grid? Answer: The area is 8 square centimetres (8 sq cm).
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Find the perimeter of a square with side 5 cm. Answer: Perimeter = 5 + 5 + 5 + 5 = 20 cm.
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Two rectangles have the same perimeter of 20 cm. Give one possible pair of lengths and breadths. Answer: Rectangle A: 6 cm by 4 cm (perimeter = 20 cm). Rectangle B: 7 cm by 3 cm (perimeter = 20 cm). Many correct answers possible.
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How many tangram pieces are there? Name them. Answer: Seven pieces — 2 large triangles, 1 medium triangle, 2 small triangles, 1 square, 1 parallelogram.
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A leaf drawn on a grid covers 10 full squares, 6 half squares, and 2 big squares (more than half). Find the approximate area. Answer: Area ≈ 10 + (6/2) + 2 = 10 + 3 + 2 = 15 sq cm.
12. Quick revision
- Area counts unit squares inside a shape.
- Perimeter is the total length around a shape.
- Use grid paper to find area by counting.
- Different shapes can share the same perimeter or area.
- Tangrams help understand area conservation.
- Practise by drawing shapes on grid paper and counting.
- Always write the correct unit — sq cm for area, cm for perimeter.
