Perimeter of a rectangle and a square
A farmer puts a fence around his rectangular field. The length of fence he needs is the distance all the way around the field. This distance is the perimeter.
The sum of the lengths of all the sides of a closed figure is its perimeter (परिमिती). It is a length, so it is measured in cm, m or km.
Rule 1: Rectangle and square — Rectangle: length + breadth + length + breadth = 2 × (length + breadth). A card 9.5 cm by 6 cm has perimeter 2 × (9.5 + 6) = 31 cm. Square: all four sides are equal, so perimeter = 4 × side. A square garden of side 15 m has perimeter 60 m.
In the figure : In the figure the perimeter of a rectangle.
- Easy : Easy: A rectangle 12 cm by 5 cm has perimeter 2 × (12 + 5) = 34 cm.
- Medium : Medium: A square of side 9 m has perimeter 4 × 9 = 36 m.
- Exam-type : Exam: A 15 m square garden fenced with 3 layers of wire needs 3 × 60 = 180 m of wire.
Rule 2: Finding a missing side — If the perimeter is given, work backwards. A square with perimeter 48 m has side 48 ÷ 4 = 12 m. A rectangle with perimeter 30 cm and length 9 cm has length + breadth = 15, so breadth = 15 − 9 = 6 cm.
- Easy : Easy: A square with perimeter 40 cm has side 10 cm.
- Medium : Medium: Perimeter 30 cm, length 9 cm gives breadth 6 cm.
- Exam-type : Exam: A rectangle's length is twice its breadth and perimeter is 36 cm. Then 2 × (2b + b) = 36, 6b = 36, b = 6 cm and l = 12 cm.
Perimeter of a rectangle and a square
In the app you can move this figure with your finger.
Open in the appPerimeter is a length. Rectangle 2(l + b). Square 4 × side.
Writing l + b as the perimeter of a rectangle. It must be doubled.
Memory clue: "आयत: दोन गुणिले (लांबी + रुंदी); चौरस: चार गुणिले बाजू".
Spot the difference
| Figure | Perimeter |
|---|---|
| Rectangle | 2 × (l + b) |
| Square | 4 × side |
Practice questions
- 1Find the perimeter of a rectangle 10 cm by 4 cm.
- 2Find the perimeter of a square of side 7 cm.
- 3In which units is perimeter measured?
- 4Find the perimeter of a field 15 m by 8 m.
- 5A square has perimeter 60 m. Find its side.
- 6A rectangle has perimeter 26 cm and length 8 cm. Find the breadth.
- 7A field 20 m by 12 m is fenced with 3 rounds of wire. How much wire is needed?
- 8Why is perimeter not measured in sq. cm?
Show answers (try them yourself first)
- 12 × 14 = 28 cm.
- 228 cm.
- 3Units of length: cm, m, km.
- 42 × 23 = 46 m.
- 560 ÷ 4 = 15 m.
- 613 − 8 = 5 cm.
- 7Perimeter 64 m, so 3 × 64 = 192 m.
- 8It is a length, not a surface.
Perimeter of a triangle
A pizza box lid, a triangular flag and a road sign all have three sides. To put a border of ribbon on a triangular flag, we need its perimeter.
Perimeter of a triangle = the sum of its three sides.
Rule 1: Perimeter of a triangle — For sides 5 cm, 7 cm and 9 cm, the perimeter is 5 + 7 + 9 = 21 cm. An equilateral triangle of side 8 cm has perimeter 3 × 8 = 24 cm.
In the figure : In the figure the perimeter of a triangle and a reshaped wire.
- Easy : Easy: Sides 3 cm, 4 cm, 5 cm give 3 + 4 + 5 = 12 cm.
- Medium : Medium: An isosceles triangle with equal sides 6 cm and base 4 cm has perimeter 6 + 6 + 4 = 16 cm.
- Exam-type : Exam: Perimeter 36 cm with two sides 11 cm and 14 cm: third side = 36 − (11 + 14) = 11 cm.
Rule 2: Same wire, different shapes — A wire can be bent into different shapes with the same perimeter. A 6 cm by 4 cm rectangle uses 20 cm of wire, so a square made from it has side 20 ÷ 4 = 5 cm. Convert all sides to the same unit before adding: 1 m + 60 cm + 90 cm = 100 + 60 + 90 = 250 cm.
- Easy : Easy: A wire of 20 cm bent into a square gives side 5 cm.
- Medium : Medium: A wire of 24 cm bent into an equilateral triangle gives side 24 ÷ 3 = 8 cm.
- Exam-type : Exam: Sides 1 m, 60 cm and 90 cm: 100 + 60 + 90 = 250 cm = 2.5 m.
Triangle perimeter = a + b + c. Same units. Same wire = same perimeter.
Adding cm and m without converting. Convert to one unit first.
Memory clue: "तीन बाजू जोडा; एकक एकच ठेवा".
Spot the difference
| Shape | Perimeter |
|---|---|
| Triangle | a + b + c |
| Equilateral triangle | 3 × side |
Practice questions
- 1Find the perimeter of a triangle with sides 6, 8 and 10 cm.
- 2Find the perimeter of an equilateral triangle of side 9 cm.
- 3The perimeter is 40 cm and two sides are 15 cm and 12 cm. Find the third side.
- 4An isosceles triangle has equal sides 7 cm and base 5 cm. Find the perimeter.
- 5A wire of 36 cm is bent into a square. Find its side.
- 6Find the perimeter of a triangle with sides 1 m, 60 cm and 90 cm in cm.
- 7A rectangle 9 cm by 3 cm is made from wire and re-bent into a square. Find the side.
- 8A wire of 30 cm is bent into an equilateral triangle. Find its side.
Show answers (try them yourself first)
- 124 cm.
- 227 cm.
- 340 − 27 = 13 cm.
- 47 + 7 + 5 = 19 cm.
- 59 cm.
- 6250 cm.
- 7Perimeter 24 cm, so side 6 cm.
- 810 cm.
Area of a rectangle and a square
To tile a floor we must know how much floor there is to cover. The amount of flat space inside a figure is called area.
Area (क्षेत्रफळ) is the space covered by a flat figure. We count unit squares. Unit: sq. cm, sq. m, sq. km.
Rule 1: Area of a rectangle and a square — A rectangle of 5 cm by 3 cm holds 5 × 3 = 15 squares of 1 sq. cm. Rectangle: area = length × breadth. Square: area = side × side. A 9 cm by 6 cm rectangle has area 54 sq. cm and a 7 cm square has area 49 sq. cm.
In the figure : In the figure area counts unit squares.
- Easy : Easy: A 6 cm by 4 cm rectangle: 6 × 4 = 24 sq. cm.
- Medium : Medium: A square of side 8 cm: 8 × 8 = 64 sq. cm.
- Exam-type : Exam: A floor 8 m by 5 m needs 8 × 5 = 40 tiles of 1 sq. m.
Rule 2: Units and missing side — Length in cm gives area in sq. cm; m gives sq. m; km gives sq. km. If the area and one side are given, divide: area 72 sq. cm and length 9 cm gives breadth 72 ÷ 9 = 8 cm.
- Easy : Easy: A square of side 5 m has area 25 sq. m.
- Medium : Medium: Area 72 sq. cm, length 9 cm gives breadth 8 cm.
- Exam-type : Exam: A floor 12 m by 9 m uses 108 tiles of 1 sq. m. At ₹ 50 per tile the cost is 108 × 50 = ₹ 5,400.
Area of a rectangle and a square
In the app you can move this figure with your finger.
Open in the appArea = l × b. Square = side × side. Unit: square units.
Writing the area in cm instead of sq. cm.
Memory clue: "क्षेत्रफळ = लांबी गुणिले रुंदी, एकक चौरस".
Spot the difference
| Figure | Area |
|---|---|
| Rectangle | l × b |
| Square | side × side |
Practice questions
- 1Find the area of a rectangle 7 cm by 3 cm.
- 2Find the area of a square of side 8 cm.
- 3In which unit is area measured, if the sides are in cm?
- 4Find the area of a field 15 m by 10 m.
- 5A rectangle has area 96 sq. cm and breadth 8 cm. Find the length.
- 6A square has area 81 sq. cm. Find its side.
- 7How many 1 sq. m tiles cover a floor 10 m by 6 m?
- 8Why is area written in square units?
Show answers (try them yourself first)
- 121 sq. cm.
- 264 sq. cm.
- 3Square centimetres (sq. cm).
- 4150 sq. m.
- 596 ÷ 8 = 12 cm.
- 69 cm.
- 760.
- 8Because we count unit squares that cover the surface.
Area of a triangle
Cut a rectangular paper along its diagonal. You get two equal triangles, so each triangle is half of the rectangle. This explains the area of a triangle.
Area of a triangle = (base × height) ÷ 2. The height (उंची) is the perpendicular from the opposite vertex to the base.
Rule 1: Area of a triangle — For base 10 cm and height 6 cm, area = (10 × 6) ÷ 2 = 30 sq. cm.
In the figure : In the figure the area of a triangle.
- Easy : Easy: Base 6 cm, height 4 cm: (6 × 4) ÷ 2 = 12 sq. cm.
- Medium : Medium: Base 14 cm, height 10 cm: 140 ÷ 2 = 70 sq. cm.
- Exam-type : Exam: A rectangle 8 cm by 5 cm cut along a diagonal gives two triangles, each of area 40 ÷ 2 = 20 sq. cm.
Rule 2: Right triangles and finding the height — In a right triangle the two sides forming the right angle are the base and the height. Legs 5 cm and 12 cm give (5 × 12) ÷ 2 = 30 sq. cm. If the area and the base are known: height = 2 × area ÷ base.
- Easy : Easy: Legs 6 cm and 8 cm give (6 × 8) ÷ 2 = 24 sq. cm.
- Medium : Medium: Legs 5 cm and 12 cm give 30 sq. cm.
- Exam-type : Exam: Area 36 sq. cm and base 9 cm: height = 2 × 36 ÷ 9 = 8 cm.
Area of a triangle
In the app you can move this figure with your finger.
Open in the appArea = (base × height) ÷ 2. Height is perpendicular to the base.
Using the slanting side as the height. The height is the perpendicular.
Memory clue: "पाया गुणिले उंची, भागिले दोन".
Spot the difference
| Triangle | Area |
|---|---|
| Any triangle | b × h ÷ 2 |
| Right triangle | leg × leg ÷ 2 |
Practice questions
- 1Find the area: base 6 cm, height 5 cm.
- 2Find the area: base 12 cm, height 7 cm.
- 3A rectangle 9 cm by 4 cm is cut along a diagonal. Find the area of each triangle.
- 4Find the area of a right triangle with legs 5 cm and 12 cm.
- 5A triangle has area 28 sq. cm and base 7 cm. Find the height.
- 6A triangle has base 14 cm and height 10 cm. Find the area.
- 7A triangle has base 8 cm and area 20 sq. cm. Find the height.
- 8Why do we divide by 2 in the area of a triangle?
Show answers (try them yourself first)
- 115 sq. cm.
- 242 sq. cm.
- 336 ÷ 2 = 18 sq. cm.
- 430 sq. cm.
- 52 × 28 ÷ 7 = 8 cm.
- 670 sq. cm.
- 740 ÷ 8 = 5 cm.
- 8A triangle is half of the rectangle with the same base and height.
Perimeter and area problems
Painting a wall needs area; putting a border around it needs perimeter. Choosing the right one is the key to solving problems.
Perimeter is a length (cm, m). Area covers a surface (sq. cm, sq. m).
Rule 1: Perimeter, area and the uncovered part — A 7 cm by 3 cm rectangle has perimeter 2 × 10 = 20 cm and area 21 sq. cm. Uncovered area = whole area − covered area. A hall 8 m by 6 m with a square carpet of side 4 m leaves 48 − 16 = 32 sq. m.
In the figure : In the figure the area not covered by the carpet.
- Easy : Easy: 9 cm by 4 cm: perimeter 26 cm and area 36 sq. cm.
- Medium : Medium: A hall 10 m by 8 m and a 5 m square carpet leave 80 − 25 = 55 sq. m.
- Exam-type : Exam: Painting a wall 5 m by 3 m at ₹ 20 per sq. m costs 15 × 20 = ₹ 300.
Rule 2: Same perimeter, different area — Rectangles 6 by 4 and 9 by 1 both have perimeter 20 cm, but areas 24 and 9 sq. cm. Doubling the side of a square makes the area 4 times: side 3 gives 9, side 6 gives 36.
- Easy : Easy: Squares of side 3 and 6: areas 9 and 36 (4 times).
- Medium : Medium: Perimeter 20 cm: 7 by 3 has area 21 and 5 by 5 has area 25.
- Exam-type : Exam: For a fixed perimeter, the square gives the largest area among rectangles.
Border → perimeter. Cover → area. Uncovered = whole − covered.
Adding perimeter and area. They are different kinds of measures.
Memory clue: "किनार = परिमिती; रंग/फरशी = क्षेत्रफळ".
Spot the difference
| Task | Use |
|---|---|
| Fencing | Perimeter |
| Tiling, painting | Area |
Practice questions
- 1Find the perimeter and area of a 9 cm by 4 cm rectangle.
- 2Find the perimeter and area of a square of side 6 cm.
- 3A hall 10 m by 8 m has a 5 m square carpet. Find the uncovered area.
- 4The side of a square is 4 cm. If it is doubled, what is the new area?
- 5Perimeter 20 cm: compare the areas of 7 by 3 and 5 by 5.
- 6Painting a wall 5 m by 3 m costs ₹ 20 per sq. m. Find the cost.
- 7A field 40 m by 25 m is fenced all around. How much fence is needed?
- 8Do equal perimeters mean equal areas? Give an example.
Show answers (try them yourself first)
- 126 cm and 36 sq. cm.
- 224 cm and 36 sq. cm.
- 380 − 25 = 55 sq. m.
- 4Side 8 cm, area 64 sq. cm, which is 4 times 16.
- 521 and 25 sq. cm; the square has the larger area.
- 615 × 20 = ₹ 300.
- 7Perimeter 2 × 65 = 130 m.
- 8No. 6 by 4 and 9 by 1 both have perimeter 20 cm but areas 24 and 9 sq. cm.