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CBSE · Class 5 · Math Mela

Chapter 11: Grandmother's Quilt (Area and Perimeter)

6 concepts · 48 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Perimeter: the length of the border

From daily life

Grandmother wants to sew lace all round the border of a quilt. The length of lace needed is the distance all the way round: the perimeter.

Simple definition

The perimeter of a shape is the total length of its border. Add the lengths of all its sides. For a rectangle opposite sides are equal; a square has 4 equal sides.

Formula: Rectangle: P = 2 × (length + breadth). Square: P = 4 × side. Any shape: add all sides.

121288Quilt12 + 8 + 12 + 8 = 40 units

In the figure: A quilt 12 units long and 8 units wide: 12 + 8 + 12 + 8 = 40 units of lace.

Rule 1: Rectangle and square — Add length and breadth, then double. For a square multiply one side by 4.

Live figure

Perimeter: the length of the border

In the app you can move this figure with your finger.

Open in the app
  • Easy: Rectangle 12 cm by 7 cm: 2 × (12 + 7) = 38 cm.
  • Medium: Square of side 9 cm: 4 × 9 = 36 cm.
  • Exam-type: Two laces: 2 long sides of 14 cm need 28 cm of red; 2 short sides of 9 cm need 18 cm of green.

Rule 2: Other shapes — Add all the sides. For a regular shape with n equal sides, multiply n by one side.

  • Easy: Triangle 6, 8 and 10 cm: 6 + 8 + 10 = 24 cm.
  • Medium: Regular pentagon, side 6 cm: 5 × 6 = 30 cm.
  • Exam-type: Regular hexagon, side 5 cm: 6 × 5 = 30 cm.
Remember

Perimeter is a length, so it is measured in cm, m or km.

Common mistake

Adding only length and breadth and forgetting to double, or adding all 4 numbers and then multiplying by 2.

Shortcut

Walk round the border and add every side once. Memory clue: "Perimeter: walk all round."

Spot the difference

ShapePerimeter
Rectangle2 × (l + b)
Square4 × s
Trianglea + b + c
Regular n-gonn × s

Practice questions

Easy
  1. 1Perimeter of a square of side 7 cm?
  2. 2Perimeter of a rectangle 9 cm by 6 cm?
  3. 3Perimeter of a triangle with sides 5, 7 and 9 cm?
Medium
  1. 4Perimeter of a regular hexagon of side 4 cm?
  2. 5A quilt is 15 units by 10 units. How much lace covers its whole border?
  3. 6Red lace on the 2 long sides and green lace on the 2 short sides of a 16 by 9 quilt. How much of each?
Challenging
  1. 7Give two pairs of length and breadth for a rectangle of perimeter 30 cm.
  2. 8A runner goes 3 times round a square park of side 25 m. How far does the runner go?
Show answers (try them yourself first)
  1. 128 cm
  2. 230 cm
  3. 321 cm
  4. 424 cm
  5. 550 units
  6. 6Red 32, green 18
  7. 710 and 5; 9 and 6 (any pair that adds to 15)
  8. 84 × 25 = 100 m; 3 rounds: 300 m

Area by tiling and counting squares

From daily life

To cover a table top you can lay tiles side by side with no gaps and no overlaps. The surface the tiles cover is the area. Counting the tiles tells us how big it is.

Simple definition

Area is the amount of flat surface a shape covers. Shapes that tile (squares, rectangles, triangles) can measure it; circles leave gaps. A square of side 1 unit has an area of 1 square unit, the standard unit of area.

Formula: Area = number of unit squares covered. Half squares: 2 halves = 1 square.

Triangles tileSquares tileCircles leave gaps

11 unit1 square unit5 units5 × 3 = 15 square units

In the figure: Triangles and squares tile with no gaps; circles leave gaps. A unit square is 1 square unit; 5 columns and 3 rows of unit squares make 15 square units.

Rule 1: Count the squares — Count the unit squares row by row, or multiply rows by columns. For a curved shape count full squares and join halves.

  • Easy: 5 rows of 7 patches: 5 × 7 = 35.
  • Medium: Leaf with 14 full and 6 half squares: 14 + 3 = 17 square units.
  • Exam-type: Garden A 18 squares, Garden B 15 squares: A is larger.

Rule 2: Choose the unit — Use square cm for small things, square m for floors and fields. 2 same triangles can make 1 square, so 14 triangles cover as much as 7 squares.

  • Easy: Postage stamp: square cm.
  • Medium: Classroom floor: square m.
  • Exam-type: Notebooks tile a table with no gaps because rectangles tile.
Remember

Area is measured in square units: square cm, square m. Compare areas only when the unit is the same.

Common mistake

Writing area in cm instead of square cm, or counting half squares as whole squares.

Shortcut

Rows × columns when the shape is a rectangle. Memory clue: "Area: cover it with squares."

Spot the difference

ShapeTiles?
SquareYes
RectangleYes
TriangleYes
CircleNo — gaps

Practice questions

Easy
  1. 1A mat has 7 rows of 8 patches. How many patches?
  2. 2Which shape leaves gaps when covering a table?
  3. 3What is the area of a square of side 1 unit?
Medium
  1. 4Garden A covers 22 squares and Garden B 19 squares. Which is larger?
  2. 5A leaf covers 12 full squares and 6 half squares. Approximate area?
  3. 6Which unit suits the area of a classroom floor?
Challenging
  1. 718 triangles cover a desk and 2 triangles make 1 square. How many squares would cover it?
  2. 85 rows of 9 notebooks cover one table. How many notebooks cover 2 such tables?
Show answers (try them yourself first)
  1. 156
  2. 2Circle
  3. 31 square unit
  4. 4Garden A
  5. 515 square units
  6. 6square m
  7. 79
  8. 85 × 9 × 2 = 90

Area of a rectangle and a square

From daily life

A patchwork has 6 rows with 4 patches in each row. Do not count each patch: 6 × 4 = 24. Rows times columns is the same as length times breadth.

Simple definition

A rectangle can be tiled by unit squares in rows and columns, so its area is length × breadth. A square is a rectangle with equal sides, so its area is side × side.

Formula: Area of rectangle = length × breadth. Area of square = side × side. Unit: square cm, square m.

646 × 4= 2424 square units

In the figure: 6 columns and 4 rows of unit squares: 6 × 4 = 24 square units.

Rule 1: Rectangle — Multiply the length by the breadth. Both must be in the same unit.

Live figure

Area of a rectangle and a square

In the app you can move this figure with your finger.

Open in the app
  • Easy: 8 cm by 5 cm: 8 × 5 = 40 square cm.
  • Medium: Field 42 m by 30 m: 42 × 30 = 1,260 square m.
  • Exam-type: Rug 3 m by 2 m: 6 square m.

Rule 2: Square — Multiply the side by itself.

  • Easy: Side 6 m: 6 × 6 = 36 square m.
  • Medium: Side 12 m: 12 × 12 = 144 square m.
  • Exam-type: Side 5 cm: area 25 square cm and perimeter 20 cm.
Remember

Area is in square units, perimeter is in plain length units. Do not mix them up.

Common mistake

Using 2 × (l + b) for area, or writing 12 × 12 = 24 for a square.

Shortcut

Area: rows × columns. Perimeter: walk round. Memory clue: "Area multiplies, perimeter adds."

Spot the difference

ShapeAreaPerimeter
Rectangle 8 by 54026
Rectangle 12 by 78438
Square 63624

Practice questions

Easy
  1. 1Area of a rectangle 11 cm by 6 cm?
  2. 2Area of a rectangle 14 cm by 9 cm?
  3. 3Area of a square of side 13 cm?
Medium
  1. 4Area of a field 25 m by 18 m?
  2. 56 rows of 4 patches: how many patches in all?
  3. 6Area of a rug 4 m by 3 m?
Challenging
  1. 7Area of a rectangle 36 cm by 25 cm?
  2. 8A floor 12 m by 9 m is tiled with 1 square m tiles at ₹60 each. What is the cost?
Show answers (try them yourself first)
  1. 166 square cm
  2. 2126 square cm
  3. 3169 square cm
  4. 4450 square m
  5. 524
  6. 612 square m
  7. 7900 square cm
  8. 812 × 9 = 108 tiles; 108 × 60 = ₹6,480

Finding a missing side

From daily life

A garden has an area of 64 square m and a length of 16 m. How wide is it? Division undoes multiplication: 64 ÷ 16 = 4 m.

Simple definition

If the area and one side of a rectangle are known, divide to find the other side. If the perimeter and one side are known, find length + breadth first (half the perimeter) and subtract.

Formula: Breadth = area ÷ length. Length = (perimeter ÷ 2) − breadth. Square: side = perimeter ÷ 4; side × side = area.

646 × 4= 2424 square units

In the figure: If 6 × 4 = 24, then 24 ÷ 6 = 4 and 24 ÷ 4 = 6: area divided by one side gives the other side.

Rule 1: From the area — Divide the area by the known side.

  • Easy: Area 144 sq m, length 12 m: breadth 144 ÷ 12 = 12 m.
  • Medium: Area 210 sq m, length 15 m: breadth 14 m.
  • Exam-type: Square of area 81 sq cm: side 9 cm (9 × 9 = 81).

Rule 2: From the perimeter — Halve the perimeter to get length + breadth, then subtract the known side.

  • Easy: Perimeter 36 cm, breadth 7 cm: length 18 − 7 = 11 cm.
  • Medium: Perimeter 50 cm, breadth 10 cm: length 25 − 10 = 15 cm.
  • Exam-type: Square of perimeter 48 cm: side 48 ÷ 4 = 12 cm.
Remember

Check by multiplying back (area) or adding back (perimeter).

Common mistake

Dividing the perimeter by the side, or forgetting to halve the perimeter before subtracting.

Shortcut

Area gives ÷, perimeter gives half then −. Memory clue: "Area: divide. Perimeter: half, then take away."

Spot the difference

KnownFindHow
Area, lengthBreadthA ÷ l
Perimeter, breadthLengthP ÷ 2 − b
Perimeter of squareSideP ÷ 4

Practice questions

Easy
  1. 1Area 96 sq m, length 12 m. Breadth?
  2. 2Area 150 sq m, length 15 m. Breadth?
  3. 3A square has perimeter 40 cm. Side?
Medium
  1. 4A square has area 64 sq cm. Side?
  2. 5Perimeter 44 cm, breadth 8 cm. Length?
  3. 6Area 245 sq cm, breadth 7 cm. Length?
Challenging
  1. 7A square has area 144 sq cm. What is its perimeter?
  2. 8A rectangle has perimeter 56 cm and length 18 cm. What is its area?
Show answers (try them yourself first)
  1. 18 m
  2. 210 m
  3. 310 cm
  4. 48 cm
  5. 514 cm
  6. 635 cm
  7. 7Side 12; perimeter 48 cm
  8. 8Breadth 28 − 18 = 10; area 180 sq cm

Same area, different perimeter and compound shapes

From daily life

A long thin rug and a squarish rug can use the same amount of cloth but need different lace around the edge. Area and perimeter are different ideas, and one does not decide the other.

Simple definition

Shapes with the same area can have different perimeters, and shapes with the same perimeter can have different areas. For a fixed area the shape closest to a square has the smallest perimeter. A compound shape (like an L) is split into rectangles to find its area.

Formula: Area 12: 1 × 12 (P = 26), 2 × 6 (P = 16), 3 × 4 (P = 14). L-shape area = sum of its rectangles.

1 × 12 P = 262 × 6 P = 163 × 4 P = 14Area 12for all three

54835 × 4 = 208 × 3 = 2420 + 24 = 44

In the figure: Top: three rectangles with area 12 but perimeters 26, 16 and 14. Bottom: an L-shape split into 5 × 4 and 8 × 3 rectangles gives 20 + 24 = 44.

Rule 1: Same area, different perimeter — List the length-and-breadth pairs that multiply to the area and find each perimeter.

Live figure

Same area, different perimeter and compound shapes

In the app you can move this figure with your finger.

Open in the app
  • Easy: Area 12: perimeters 26, 16, 14.
  • Medium: Area 18: 1 × 18 → 38, 2 × 9 → 22, 3 × 6 → 18.
  • Exam-type: Perimeter 20: 9 × 1 → area 9, 7 × 3 → 21, 5 × 5 → 25 (largest).

Rule 2: Compound shapes — Split the shape into rectangles and add their areas, or cut a corner out of a big rectangle and subtract. Cutting a corner out leaves the perimeter the same as the big rectangle.

  • Easy: L-shape: 8 × 3 + 5 × 4 = 24 + 20 = 44 square cm.
  • Medium: 10 × 6 with a 4 × 3 corner cut: 60 − 12 = 48 square cm.
  • Exam-type: Its perimeter is still 2 × (10 + 6) = 32 cm.
Remember

Compare both area and perimeter before judging two shapes; one number alone does not tell the other.

Common mistake

Believing "bigger area means bigger perimeter" (1 × 12 has area 12 and perimeter 26; 4 × 4 has area 16 and perimeter 16).

Shortcut

Try every pair of factors; the pair closest together gives the smallest perimeter. Memory clue: "Squarish shapes save lace."

Spot the difference

RectangleAreaPerimeter
1 × 121226
2 × 61216
3 × 41214
5 × 52520

Practice questions

Easy
  1. 1Area 18: which rectangle has the smallest perimeter: 1 × 18, 2 × 9 or 3 × 6?
  2. 2Perimeter of a 1 × 18 rectangle?
  3. 3Perimeter 16: 4 × 4, 5 × 3 or 7 × 1. Which has the largest area?
Medium
  1. 4True or false: shapes with the same area always have the same perimeter.
  2. 5An L-shape is a 6 × 2 rectangle joined to a 3 × 3 rectangle. Total area?
  3. 6A 12 × 8 rectangle has a 5 × 4 corner cut out. Area left?
Challenging
  1. 7What is the perimeter of that same shape?
  2. 8Six 1 cm squares form a 2 × 3 block or a row of 6. Perimeter of each?
Show answers (try them yourself first)
  1. 13 × 6
  2. 238 units
  3. 34 × 4 (area 16)
  4. 4False
  5. 512 + 9 = 21 square units
  6. 696 − 20 = 76 square units
  7. 72 × (12 + 8) = 40 units
  8. 8Block 10 cm; row 14 cm

Area and perimeter word problems

From daily life

Fencing a field, laying tiles, buying a carpet and edging a quilt are everyday area-and-perimeter jobs. First decide: is it the border (perimeter) or the surface (area)?

Simple definition

Border jobs (fence, lace, rope, boundary) use perimeter. Surface jobs (tiles, carpet, paint, ground cover) use area. Then use the price or size of one unit to finish.

Formula: Fence cost = perimeter × rate per m. Tiles needed = floor area ÷ area of one tile. Carpet cost = area × rate per sq m.

121288Quilt12 + 8 + 12 + 8 = 40 units

646 × 4= 2424 square units

In the figure: Top: a border job uses the perimeter. Bottom: a tiling job uses the area, rows times columns.

Rule 1: Border or surface? — Read what is to be covered: a line along the edge needs perimeter, a flat region needs area.

  • Easy: Field 40 m by 25 m, fence: 2 × (40 + 25) = 130 m; at ₹12 a metre ₹1,560.
  • Medium: Rope 36 m bent into a square: side 36 ÷ 4 = 9 m.
  • Exam-type: Quilt 14 by 10, red lace 30: border 48, green lace 48 − 30 = 18.

Rule 2: Tiles and carpets — Find the area of the floor, then divide by the area of one tile, or multiply by the price per square unit.

  • Easy: Floor 8 m by 6 m, tiles 1 m side: 48 tiles; tiles 2 m side: 48 ÷ 4 = 12 tiles.
  • Medium: Carpet 6 m by 4 m at ₹50 per sq m: 24 × 50 = ₹1,200.
  • Exam-type: Quilt 120 by 90 cm, 10 cm patches: 12 × 9 = 108 patches.
Remember

Bigger tiles mean fewer tiles: halve each side and you need 4 times as many.

Common mistake

Using perimeter to count tiles, or area to count fence.

Shortcut

Underline "round", "border", "fence" for perimeter; "cover", "tile", "carpet" for area. Memory clue: "Edge = perimeter, cover = area."

Spot the difference

JobUse
Fence a fieldperimeter
Lace on a quiltperimeter
Tile a floorarea
Buy a carpetarea

Practice questions

Easy
  1. 1A floor 10 m by 7 m is tiled with 1 m square tiles. How many tiles?
  2. 2A field 60 m by 40 m is fenced. How long is the fence?
  3. 3Fencing costs ₹15 per metre. Cost for that 200 m?
Medium
  1. 4A carpet 5 m by 3 m costs ₹80 per sq m. Total cost?
  2. 5A 48 m rope is bent into a square. Side?
  3. 6A quilt 150 cm by 100 cm has 10 cm square patches. How many patches?
Challenging
  1. 7A 20 by 15 quilt needs red lace 40 and the rest green. How much green?
  2. 8A 30 m by 10 m field and a 20 m square field both need 80 m of fence. Which has more area, and by how much?
Show answers (try them yourself first)
  1. 170
  2. 2200 m
  3. 3₹3,000
  4. 415 × 80 = ₹1,200
  5. 512 m
  6. 615 × 10 = 150
  7. 7Border 70; green 30
  8. 8Square: 400 vs 300, by 100 sq m