NCERT Solutions for Class 5th Maths Chapter 11 Counting patches; which shapes tile a table top; meeting the word area — The Rug and the Four Tables
Book page 143 Updated on2026-09-19
Q1.
Preetha and Adrit's grandmother is making a rug with square patches. The picture below shows the rug. How many patches have they used to make this?
Grandmother's rug. Every patch is a square of cloth.
Answer
90 patches.
You do not have to count all 90. Count one row and one column, then multiply.
Count along the top. There are 15 patches in one row.
Count down one side. There are 6 rows.
Multiply. Every row has the same 15 patches, and there are 6 such rows.
Patches = 15 × 6 Patches = 90 patches
Why it happens: The patches are laid out in neat rows and columns. When things are set out this way you never need to count them one by one. Counting one row and one column is enough, because 6 rows of 15 is just 15 taken 6 times.
Check it yourself: Add instead of multiplying. 15 + 15 + 15 + 15 + 15 + 15 = 90. Same answer.
Q2.
They found that __________, __________ and __________ shapes cover the top of the table without gaps and overlaps. __________ shape leaves gaps.
The four tables. Preetha used triangles and circles; Adrit used squares and rectangles.
Answer
Triangles, squares and rectangles cover the top of the table without gaps and overlaps. The circle shape leaves gaps.
Table
Tile used
Any gaps?
Table 1
Triangles
No gaps
Table 2
Circles
Gaps left
Table 3
Squares
No gaps
Table 4
Rectangles
No gaps
Round tiles always leave little empty corners. They do not tile.
Why it happens: Triangles, squares and rectangles have straight edges. A straight edge can lie flat against the next straight edge with nothing left between them. A circle has a curved edge. Two curves touch at only one point, so small empty corners are always left over.
Word to remember: When shapes fit together with no gaps and no overlaps, we say they tile the region. Look at the floor tiles in your school — they are squares or rectangles, never circles.
Q3.
__________ triangles cover Table 1.
Answer
20 triangles cover Table 1.
Every small square holds 2 triangles. 10 squares × 2 = 20 triangles.
Look at the pattern. The table top is first divided into small squares — 5 across and 2 down.
Count the squares. 5 × 2 = 10 squares.
Look inside one square. A line runs from corner to corner, cutting it into 2 triangles.
Multiply. 10 squares × 2 triangles = 20.
Squares on the table top = 5 × 2 = 10 Triangles in each square = 2 Triangles in all = 10 × 2 = 20 triangles
Why it happens: Cutting a square along its corner-to-corner line makes two triangles of exactly the same size. Nothing is thrown away, so every square turns into two triangles.
Q4.
__________ squares cover Table 3.
Answer
8 squares cover Table 3.
The thick black lines mark the square tiles: 4 × 2 = 8.
Count along the top. 4 squares.
Count down the side. 2 rows.
Multiply. 4 × 2 = 8.
Squares = 4 × 2 = 8 squares
Look carefully: The green cloth has a small check pattern printed on it. Do not count the tiny printed checks. Count only the tiles marked by the thick dividing lines.
Q5.
__________ rectangles cover Table 4.
Answer
12 rectangles cover Table 4.
3 × 4 = 12 rectangular tiles.
Count along the top. 3 rectangles.
Count down the side. 4 rows.
Multiply. 3 × 4 = 12.
Rectangles = 3 × 4 = 12 rectangles
Why the three tables give three different numbers: The table tops are the same size, but the tiles are not. Table 3 uses big squares, so only 8 are needed. Table 4 uses smaller rectangles, so 12 are needed. Table 1 uses even smaller triangles, so 20 are needed. Smaller tile, bigger number.
Q6.
Do circles tile? Can we use them to cover a region?
Answer
No, circles do not tile. We cannot use them to cover a region completely.
Round tiles always leave little empty corners. They do not tile.
Push two circles together. They touch at only one point.
Look at the space around that point. A small curved corner is left empty on every side.
Try to fill that corner. Another circle will not fit into it, because the corner is not round.
Why it happens: To cover a region with no gaps, the edge of one tile must lie flat along the edge of the next. Only straight edges can do that. A circle is curved everywhere, so it can never lie flat against its neighbour.
Try This: Put five one-rupee coins on your desk and push them as close as you can. You will still see little white gaps between them. Now do the same with five square erasers — no gaps at all.
Q7.
The area of Table 1 is __________ triangle units.
Table 1, covered with triangles.
Answer
The area of Table 1 is 20 triangle units.
The word area means the amount of surface a shape covers — the cloth you would need to cover it. We measure area by counting how many tiles fit on it. Here the tile is one triangle, so we count in triangle units.
Small squares on the table top = 5 × 2 = 10 Triangles in each square = 2 Area = 10 × 2 = 20 triangle units
Why we say “triangle units”: A number on its own does not tell us the area. “20” is meaningless until we say 20 of what. Here we chose the triangle as our tile, so the answer is 20 triangle units.
Q8.
The area of Table 3 is __________ square units.
Table 3, covered with squares.
Answer
The area of Table 3 is 8 square units.
Squares across = 4 Rows of squares = 2 Area = 4 × 2 = 8 square units
Why it happens: The whole table top is exactly filled by 8 equal squares, with nothing left over and nothing doubled up. So 8 of these squares is the amount of surface the table top has.
Tip: The square is the tile we shall use for the rest of this chapter, because all four of its sides are the same length. That makes it easy to draw and easy to count.
Q9.
The area of Table 4 is __________ rectangle units.
Table 4, covered with rectangles.
Answer
The area of Table 4 is 12 rectangle units.
Rectangles across = 3 Rows of rectangles = 4 Area = 3 × 4 = 12 rectangle units
Now put the three answers side by side.
Table
Tile chosen
Area of the same table top
Table 1
triangle
20 triangle units
Table 3
square
8 square units
Table 4
rectangle
12 rectangle units
Why the numbers are different: The table tops are the same, but each child chose a different tile. A small tile fits in more times, a big tile fits in fewer times. This is why we must always say which unit we used. Later in the chapter everybody uses the same unit — the 1 cm square — and then the numbers can be compared fairly.
Q10.
Now, try to cover the top of your table without gaps and overlaps with the following objects of same size. (a) Notebooks (b) Lunch boxes (c) Pencil boxes (d) Maths textbooks. Which of the above objects covered the region completely?
Answer
The notebooks and the maths textbooks cover the table completely. The lunch boxes and the pencil boxes leave gaps.
Object
Shape of its base
Does it cover completely?
(a) Notebooks
flat rectangle
Yes — no gaps
(b) Lunch boxes
often round or oval, with curved corners
No — gaps at the corners
(c) Pencil boxes
long and narrow, curved ends
No — gaps at the ends
(d) Maths textbooks
flat rectangle
Yes — no gaps
Lay the objects in rows. Start at one corner of the table and keep the edges touching.
Look down at the table from above. Can you see any bit of the table top?
Decide. If no table top shows, the object covers completely.
Why it happens: Only objects with straight edges and square corners can sit edge to edge with nothing between them. Notebooks and textbooks are flat rectangles, so they work. Lunch boxes and pencil boxes have rounded shapes, so little empty patches are always left, just like the circles on Table 2.
Sample answer: “We covered our table with maths textbooks. It took 3 books across and 4 books down, so 12 books covered the whole table with no gaps. When we tried the lunch boxes, small gaps were left at every corner, so they did not cover it completely.”