Fencing a Field Versus Carpeting It
Learn to calculate perimeter and area for squares, rectangles and irregular shapes, find a missing side, and solve path and cost problems without mixing units.
What is the difference between perimeter and area?
Perimeter is the total distance around the boundary of a figure, measured in units of length such as cm or m. Area is the amount of surface a figure covers, measured in square units such as or . Fencing a field uses perimeter; carpeting a room uses area.
This page covers everything in the ICSE Class 6 Mathematics chapter on perimeter and area: finding the perimeter of rectilinear figures, measuring irregular areas on squared paper, the square and rectangle formulas, and problems on paths and costs.
This page covers everything in the ICSE Class 6 Mathematics chapter on perimeter and area: finding the perimeter of rectilinear figures, measuring irregular areas on squared paper, the square and rectangle formulas, and problems on paths and costs.
Formula
How do you find the perimeter of different figures?
Perimeter is always the sum of all the sides, which simplifies for regular shapes:
Worked example. A rectangle measures 12 m by 8 m, so
A square of side 7 cm gives cm, and a regular hexagon of side 5 cm gives cm.
For a triangle or any other rectilinear figure, simply add every side.
Units must match before adding. To find the perimeter of a rectangle 2 m by 50 cm, convert first: 50 cm is 0.5 m, so m. Adding 2 and 50 directly would be meaningless.
Worked example. A rectangle measures 12 m by 8 m, so
A square of side 7 cm gives cm, and a regular hexagon of side 5 cm gives cm.
For a triangle or any other rectilinear figure, simply add every side.
Units must match before adding. To find the perimeter of a rectangle 2 m by 50 cm, convert first: 50 cm is 0.5 m, so m. Adding 2 and 50 directly would be meaningless.
How do you find the area of an irregular figure on squared paper?
When a shape has no formula, count the squares it covers using a simple convention:
- count every full square as 1
- count every square that is more than half covered as 1
- count every exactly half covered square as
- ignore squares less than half covered
Suppose a leaf traced on squared paper covers 14 full squares, 6 more-than-half squares and 4 exact halves. Then
If each square is , the area is about .
In real life this is how you would estimate the area of a curved plot on a map.
The method gives an estimate, not an exact value — so questions using it say "approximate area", and you should not present the answer as exact.
- count every full square as 1
- count every square that is more than half covered as 1
- count every exactly half covered square as
- ignore squares less than half covered
Suppose a leaf traced on squared paper covers 14 full squares, 6 more-than-half squares and 4 exact halves. Then
If each square is , the area is about .
In real life this is how you would estimate the area of a curved plot on a map.
The method gives an estimate, not an exact value — so questions using it say "approximate area", and you should not present the answer as exact.
Formula
How do you find the area of a square and a rectangle?
For these two shapes the area formulas are exact:
A square of side 9 cm has area . A rectangle 15 m by 6 m has area .
The formulas also run backwards to find a missing side. If a rectangle has area and length 12 cm, then
And if a square has area , its side is the number that multiplies by itself to give 49, so m.
Notice the units: multiplying cm by cm gives . Writing an area answer in plain cm is marked wrong even when the number is right.
A square of side 9 cm has area . A rectangle 15 m by 6 m has area .
The formulas also run backwards to find a missing side. If a rectangle has area and length 12 cm, then
And if a square has area , its side is the number that multiplies by itself to give 49, so m.
Notice the units: multiplying cm by cm gives . Writing an area answer in plain cm is marked wrong even when the number is right.
How do you solve path and cost problems?
For a path, find the area of the larger rectangle and subtract the area of the smaller one — the difference is the path.
A garden measures 20 m by 15 m with a 2 m wide path running around the outside. The outer rectangle is m by m, so
If instead the path ran inside the garden, the inner rectangle would be m by m, giving .
For cost, first decide whether the job needs perimeter or area. Fencing follows the boundary, so it uses perimeter; levelling or carpeting covers the surface, so it uses area.
Fencing that 20 m by 15 m garden at Rs 50 per metre costs m of fence, so . Carpeting it at Rs 80 per square metre costs .
The width of a path is added or subtracted twice, once at each side — the commonest slip in the whole topic.
A garden measures 20 m by 15 m with a 2 m wide path running around the outside. The outer rectangle is m by m, so
If instead the path ran inside the garden, the inner rectangle would be m by m, giving .
For cost, first decide whether the job needs perimeter or area. Fencing follows the boundary, so it uses perimeter; levelling or carpeting covers the surface, so it uses area.
Fencing that 20 m by 15 m garden at Rs 50 per metre costs m of fence, so . Carpeting it at Rs 80 per square metre costs .
The width of a path is added or subtracted twice, once at each side — the commonest slip in the whole topic.
Exam tip
The mistake most students make with path width
For a 2 m path around a garden, students add 2 to each dimension instead of 4. The path runs along both sides, so each dimension grows by twice the width.
Draw a quick sketch and mark the width at both ends before calculating. A 20 m side with a 2 m path outside becomes m.
The same doubling applies inwards: an inside path of width 2 m reduces a 20 m side to m. And always check the answer's direction — an outer path must give a larger rectangle, an inner path a smaller one.
Draw a quick sketch and mark the width at both ends before calculating. A 20 m side with a 2 m path outside becomes m.
The same doubling applies inwards: an inside path of width 2 m reduces a 20 m side to m. And always check the answer's direction — an outer path must give a larger rectangle, an inner path a smaller one.
Did you know
Why is area measured in square units?
Area counts how many unit squares fit inside a shape. A square of side 1 cm is one , and a rectangle 4 cm by 3 cm holds exactly 12 such squares.
That is why multiplying cm by cm produces — the two lengths are building a grid, and the answer counts its cells rather than a distance.
That is why multiplying cm by cm produces — the two lengths are building a grid, and the answer counts its cells rather than a distance.
Key takeaways
Perimeter and area in 30 seconds
- Perimeter is the distance around a figure in units of length; area is the surface covered, in square units.
- Perimeter of a square is and of a rectangle ; convert all measurements to the same unit before adding.
- Estimate an irregular area on squared paper by counting full squares, counting more-than-half squares as one and exact halves as a half.
- Area of a square is and of a rectangle ; divide the area by one dimension to recover the other.
- A path's area is the difference of two rectangles, and its width changes each dimension twice; fencing uses perimeter while carpeting and levelling use area.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Perimeter of a square is and of a rectangle ; convert all measurements to the same unit before adding.
- Estimate an irregular area on squared paper by counting full squares, counting more-than-half squares as one and exact halves as a half.
- Area of a square is and of a rectangle ; divide the area by one dimension to recover the other.
- A path's area is the difference of two rectangles, and its width changes each dimension twice; fencing uses perimeter while carpeting and levelling use area.
You will remember all of this far better after answering five questions on it than after reading it twice.