NCERT Solutions for Class 5th Maths Chapter 11 Same area but different perimeter; same perimeter but different area — Let Us Explore
Book page 147–148 Updated on2026-09-19
Q1.
Tick the shapes with the same area.
Six shapes drawn on a 1 cm square grid.
Answer
Tick (a), (b), (c), (d) and (f). Each of them covers 12 square cm. Only shape (e) is different — it covers 8 square cm.
Count the shaded squares in each shape. Each grid square is 1 cm by 1 cm, so each one is 1 square cm.
Use rows and columns where you can. For a rectangle, multiply instead of counting one by one.
For the staircase shape (b), count carefully. The top row has 2 squares and the five rows below have 2 squares each: 2 + 10 = 12.
Compare the six numbers and tick the ones that match.
Shape
How it is made
Area
Tick?
(a)
6 across, 2 down
12 square cm
✓
(b)
2 + (5 × 2) staircase
12 square cm
✓
(c)
3 across, 4 down
12 square cm
✓
(d)
4 across, 3 down
12 square cm
✓
(e)
8 across, 1 down
8 square cm
✗
(f)
12 across, 1 down
12 square cm
✓
Why so many different-looking shapes share one area: Area only asks how many squares, never where they are. Twelve squares in a long line, twelve in a fat block and twelve in a staircase are still twelve squares.
Q2.
Find the perimeters of these shapes.
Six shapes drawn on a 1 cm square grid.
Answer
Here are the perimeters. They are not the same, even though the areas are.
Shape
Area
Working for the perimeter
Perimeter
(a) 6 × 2
12 square cm
6 + 2 + 6 + 2
16 cm
(b) staircase
12 square cm
count the edge steps
18 cm
(c) 3 × 4
12 square cm
3 + 4 + 3 + 4
14 cm
(d) 4 × 3
12 square cm
4 + 3 + 4 + 3
14 cm
(e) 8 × 1
8 square cm
8 + 1 + 8 + 1
18 cm
(f) 12 × 1
12 square cm
12 + 1 + 12 + 1
26 cm
How to measure the perimeter of a shape drawn on a grid:
Put your pencil on one corner of the shape.
Travel along the outline and say “one” for every 1 cm side you pass.
Do not cut across the middle. Follow every step in and out.
Stop when you come back to your starting corner and read off your count.
Shape (b), the staircase: going round the edge you pass 18 sides of 1 cm each Perimeter of (b) = 18 cm
Tip for rectangles: You never need to walk round. Perimeter = 2 × (length + breadth). For (f), 2 × (12 + 1) = 2 × 13 = 26 cm.
Q3.
What do you notice? Discuss.
Answer
We notice this: shapes can have the same area and still have very different perimeters.
All of (a), (b), (c), (d) and (f) have area = 12 square cm But their perimeters are 16 cm, 18 cm, 14 cm, 14 cm and 26 cm
The smallest perimeter belongs to the shapes closest to a square — (c) 3 × 4 and (d) 4 × 3, both 14 cm.
The largest perimeter belongs to the longest, thinnest shape — (f) 12 × 1, a huge 26 cm.
The staircase (b) has extra steps in its outline, so it needs more edge than the plain rectangle (a) that holds the same 12 squares.
Why this happens: Think of 12 patches of cloth. If you stitch them into a fat block, most of the joins are hidden inside and only a short edge is left showing. If you stitch them into one long strip, almost every patch has two of its sides on the outside, so the edge is very long. Same cloth, very different amount of lace.
Sample answer for the discussion: “All five shapes need the same amount of cloth, because each is made of 12 one-centimetre squares. But they do not need the same amount of lace. The 3 × 4 shape needs only 14 cm of lace, while the 12 × 1 strip needs 26 cm. So knowing the area does not tell us the perimeter.”
Q4.
Tick the shapes with the same perimeter.
Five shapes drawn on a 1 cm square grid.
Answer
Tick (c), (d) and (e). Each of these three has a perimeter of 16 cm. Shapes (a) and (b) form a second matching pair, both 12 cm.
How to count the perimeter of a stepped shape:
Start at one corner of the outline and choose a direction.
Follow the outline all the way round, counting each 1 cm side as you pass it.
Go in and out of every step. Do not take short cuts across a corner.
Stop where you started and read your total.
Shape
Perimeter
Tick?
(a)
12 cm
— (matches (b))
(b)
12 cm
— (matches (a))
(c)
16 cm
✓
(d)
16 cm
✓
(e)
16 cm
✓
Why (a) and (b) come out shorter: Both of them are almost a filled 3 × 3 block, which is a fat, tidy shape with a short edge. Shapes (c), (d) and (e) are spread out — a plus-shape, an L and a long strip — so their outlines have more sides poking out.
Q5.
Find the areas of these shapes.
Five shapes drawn on a 1 cm square grid.
Answer
All five shapes have the same area — 7 square cm each.
Shape
How to count
Area
Perimeter
(a)
3 + 3 + 1
7 square cm
12 cm
(b)
3 + 2 + 2
7 square cm
12 cm
(c)
2 + 1 + 3 + 1
7 square cm
16 cm
(d)
4 + 3
7 square cm
16 cm
(e)
7 in one row
7 square cm
16 cm
Take one shape at a time.
Count the shaded squares row by row and write each row's count.
Add the row counts. For shape (c): 2 + 1 + 3 + 1 = 7.
Write the unit. Each grid square is 1 square cm, so the area is 7 square cm.
Why counting row by row is safer than counting one by one: With stepped shapes it is very easy to miss a square or count one twice. Taking one row at a time keeps your place, and adding four small numbers is easier than counting to seven around a corner.
Q6.
What do you notice? Discuss.
Answer
We notice this: shapes can have the same perimeter and still be different shapes — and having the same area does not mean having the same perimeter either.
All five shapes: area = 7 square cm Perimeters: (a) 12 cm, (b) 12 cm, (c) 16 cm, (d) 16 cm, (e) 16 cm
Shapes (c), (d) and (e) look completely different — a plus, an L and a straight strip — yet each needs exactly 16 cm of lace.
Shapes (a) and (b) hold the very same 7 squares but need only 12 cm of lace, because they are packed into a tidy block.
So the perimeter depends on how spread out the squares are, not on how many there are.
Put both explorations together: On page 147 we saw shapes with the same area and different perimeters. Here we see shapes with the same perimeter that look nothing alike. Area and perimeter are two separate measurements of a shape. Knowing one never tells you the other.
Sample answer for the discussion: “Shapes (c), (d) and (e) all need 16 cm of lace even though one is a plus, one is an L and one is a straight strip. All five shapes need 7 square cm of cloth. So two shapes can need the same lace and the same cloth and still not be the same shape.”