Perimeter of polygons
Hari wants to stick a coloured border round a rectangular notice board. A farmer wants to fence his field. Both first need to know the total length all the way round the shape.
The perimeter of a closed shape with straight sides is the sum of the lengths of all its sides. It is a length, so it is measured in cm, m and so on. For a rectangle it is twice the sum of its length and width, and for a square it is four times its side.
In the figure: A 6 by 4 rectangle: 6 + 4 + 6 + 4 = 20, which is also 2 × (6 + 4).
Rule 1: Add the sides — For any polygon add all its sides. A rectangle has two equal lengths and two equal widths, so its perimeter is 2 × (length + width). A square has four equal sides, so its perimeter is 4 × side.
Perimeter of polygons
In the app you can move this figure with your finger.
Open in the app- Easy: Rectangle 15 cm by 9 cm: 15 + 9 = 24 and 24 × 2 = 48 cm.
- Medium: Square of side 12 m: 12 × 4 = 48 m.
- Exam-type: Triangle with sides 7 cm, 9 cm and 11 cm: 7 + 9 = 16 and 16 + 11 = 27 cm.
Rule 2: Find a missing side — Subtract the known sides from the perimeter. For a rectangle, half the perimeter is length + width, so take the known one away from it. For a square, divide the perimeter by 4.
- Easy: Rectangle with perimeter 46 cm and length 14 cm: half is 23, and 23 − 14 = 9 cm is the width.
- Medium: Square with perimeter 64 m: 64 ÷ 4 = 16 m is one side.
- Exam-type: Triangle with perimeter 36 cm and two sides 10 cm and 13 cm: 10 + 13 = 23 and 36 − 23 = 13 cm is the third side.
Perimeter is a length, so write cm or m, never square units.
Adding just one length and one width, and forgetting to double it.
For a rectangle, first find length + width, then double it. Memory clue: “Perimeter = Path all around”.
Spot the difference
| Shape | Perimeter | Example |
|---|---|---|
| Rectangle | 2 × (length + width) | 2 × (15 + 9) = 48 |
| Square | 4 × side | 4 × 12 = 48 |
| Triangle | sum of 3 sides | 7 + 9 + 11 = 27 |
| Any polygon | sum of all sides | 8 + 11 + 9 + 14 = 42 |
Practice questions
- 1What is the perimeter of a rectangle 10 cm long and 4 cm wide?
- 2What is the perimeter of a square of side 7 m?
- 3A triangle has sides 6 cm, 8 cm and 9 cm. What is its perimeter?
- 4A rectangle has a perimeter of 60 cm and a length of 18 cm. What is its width?
- 5A square has a perimeter of 52 m. What is one side?
- 6A plot is 25 m by 15 m. Fencing costs ₹12 per metre. What is the total cost?
- 7A 56 cm wire is bent into a rectangle of length 20 cm. What is the width?
- 8A park is 80 m by 60 m. Rohan runs round it 3 times. How far does he run?
Show answers (try them yourself first)
- 11) 28 cm
- 22) 28 m
- 33) 23 cm
- 44) 12 cm
- 55) 13 m
- 66) ₹960
- 77) 8 cm
- 88) 840 m (one round is 280 m)
Regular polygons and unknown sides
A honeycomb cell is a regular hexagon and a stop sign is a regular octagon. A tailor puts lace round a pentagon-shaped cushion and needs the length of the whole edge.
A regular polygon has all sides equal and all angles equal. An equilateral triangle, a square, a regular pentagon, a regular hexagon and a regular octagon have 3, 4, 5, 6 and 8 sides. Its perimeter is the number of sides × the length of one side.
In the figure: The more sides a regular polygon has, the more times its side is added.
Rule 1: Perimeter = sides × one side — Because every side is equal, do not add them one by one. Multiply the number of sides by the length of one side.
Regular polygons and unknown sides
In the app you can move this figure with your finger.
Open in the app- Easy: Regular pentagon, side 9 cm: 5 × 9 = 45 cm.
- Medium: Regular hexagon, side 14 m: 6 × 14 = 84 m.
- Exam-type: Regular octagon, side 6 cm: 8 × 6 = 48 cm.
Rule 2: Find the side or the number of sides — Divide the perimeter by the number of sides to get one side. Divide the perimeter by one side to get the number of sides.
- Easy: Regular hexagon with perimeter 72 cm: 72 ÷ 6 = 12 cm is one side.
- Medium: Equilateral triangle with perimeter 45 m: 45 ÷ 3 = 15 m is one side.
- Exam-type: A regular polygon has perimeter 63 cm and side 9 cm: 63 ÷ 9 = 7, so it has 7 sides.
A square is a regular polygon, but a rectangle that is not a square is not, because its sides are unequal.
Calling every 5-sided or 6-sided shape regular even when its sides or angles are unequal.
Find one side with perimeter ÷ number of sides. Memory clue: “Regular = Same sides, Same angles”.
Spot the difference
| Polygon | Sides | Perimeter |
|---|---|---|
| Equilateral triangle | 3 | 3 × side |
| Square | 4 | 4 × side |
| Regular pentagon | 5 | 5 × side |
| Regular hexagon | 6 | 6 × side |
Practice questions
- 1How many sides does an octagon have?
- 2An equilateral triangle has a side of 11 cm. What is its perimeter?
- 3A regular pentagon has a side of 4 cm. What is its perimeter?
- 4A regular hexagon has a perimeter of 48 cm. What is one side?
- 5Divya has a 50 cm wire and bends a regular octagon of side 5 cm. How much wire is left?
- 6A square plot has a perimeter of 84 m. What is one side?
- 7A regular polygon has a perimeter of 96 cm and each side is 12 cm. How many sides does it have?
- 8An equilateral triangle of side 24 cm and a regular hexagon have the same perimeter. What is the side of the hexagon?
Show answers (try them yourself first)
- 11) 8
- 22) 33 cm
- 33) 20 cm
- 44) 8 cm
- 55) 10 cm (40 cm is used)
- 66) 21 m
- 77) 8 sides (an octagon)
- 88) 12 cm (perimeter 72 ÷ 6)
Area: counting unit squares and estimating
Esha wants to cover her room floor with square tiles. How many tiles she needs depends on the space inside the room, not on the length of its edge. The paint for a wall also depends on the space it covers.
The area of a closed figure is the measure of the region it encloses. We measure it by counting unit squares, so it is given in square units such as square cm or square m. To estimate the area of an irregular shape, count the full squares and count squares that are more than half filled as 1.
In the figure: 3 rows of 5 unit squares make 15 unit squares, so the area is 15 square units.
Rule 1: Count the squares in a rectangle — A rectangle made of unit squares has some rows with the same number of squares in each row. Multiply the number of rows by the squares in one row.
Area: counting unit squares and estimating
In the app you can move this figure with your finger.
Open in the app- Easy: 3 rows of 9 squares: 3 × 9 = 27 square units.
- Medium: 5 rows of 8 squares: 5 × 8 = 40 square units.
- Exam-type: A floor is 12 squares long and 7 squares wide: 12 × 7 = 84 square units.
Rule 2: Estimate an uneven shape — Trace the shape on a grid. Count each full square as 1, each square more than half filled as 1, each exactly half as half, and ignore squares less than half filled.
- Easy: 18 full, 5 more than half and 4 less than half: 18 + 5 = 23 square units (about).
- Medium: 10 full squares and 6 exactly half squares: 6 halves make 3, so 10 + 3 = 13 square units.
- Exam-type: 25 full, 8 more than half, 2 exactly half and 6 less than half: 25 + 8 = 33 and 33 + 1 = 34 square units.
Shapes made of the same number of unit squares have the same area, even if their shapes differ.
Mixing up area and perimeter, or writing cm instead of square cm.
Rows × squares in each row gives the count at once. Memory clue: “Area counts the Inside, perimeter walks the edge”.
Spot the difference
| Measure | What it tells | Unit |
|---|---|---|
| Perimeter | length of the edge | cm, m |
| Area | space inside | square cm, square m |
| Unit square | 1 by 1 square | 1 square unit |
| Estimate | count full and more-than-half | about |
Practice questions
- 1Which unit is used for area: cm or square cm?
- 2A rectangle has 6 rows of 7 unit squares. What is its area?
- 3Two different shapes are each made of 9 unit squares. What is the area of each?
- 4A figure covers 8 full squares and 6 half squares. What is its area?
- 5A floor 9 m by 6 m is covered with tiles of 1 square metre each. How many tiles are needed?
- 6Each small square of a grid is 3 cm by 3 cm. A shape covers 12 such squares. What is its area?
- 7A figure covers 30 full squares, 7 more than half, 4 less than half and 6 exactly half. Estimate its area.
- 8A rectangle of unit squares has an area of 56 square units and 7 rows. How many squares are in each row?
Show answers (try them yourself first)
- 11) square cm
- 22) 42 square units
- 33) 9 square units each
- 44) 11 square units (8 + 3)
- 55) 54 tiles
- 66) 108 square cm (12 × 9)
- 77) 40 square units (30 + 7 + 3)
- 88) 8 squares
Area of rectangles and squares
A farmer buys seeds by the area of his field, and a painter charges by the area of the wall. Both need a quick way to get the area without drawing and counting every square.
The area of a rectangle = length × width. The area of a square = side × side. The answer is in square units, such as square cm or square m. Two shapes can have the same area but different perimeters, or the same perimeter but different areas.
In the figure: Both rectangles have area 12, yet their perimeters are 16 and 14.
Rule 1: Multiply the two sides — Area of a rectangle: multiply length by width. For a square, multiply the side by itself. If the area and one side are known, divide the area by that side to get the other side.
- Easy: Rectangle 14 cm by 6 cm: 14 × 6 = 84 square cm.
- Medium: Square of side 11 m: 11 × 11 = 121 square m.
- Exam-type: Rectangle of area 96 square cm and length 12 cm: 96 ÷ 12 = 8 cm is the width.
Rule 2: Same perimeter, different area — Keep the perimeter fixed and change the length: the width changes too, and so does the area. The closer the shape is to a square, the larger its area.
Area of rectangles and squares
In the app you can move this figure with your finger.
Open in the app- Easy: Perimeter 20 with sides 9 and 1: area 9 × 1 = 9 square units.
- Medium: Perimeter 20 with sides 8 and 2: area 8 × 2 = 16 square units.
- Exam-type: Perimeter 20 with sides 5 and 5, a square: area 5 × 5 = 25 square units, the largest.
To find the area of an L-shape, split it into rectangles and add, or take a corner rectangle away from a big one.
Adding length and width to get the area, or writing square cm as cm.
Area asks “how many squares fit inside”, so multiply; perimeter asks “how far round”, so add. Memory clue: “Area = Across × Up (length × width)”.
Spot the difference
| Shape | Area | Example |
|---|---|---|
| Rectangle | length × width | 14 × 6 = 84 |
| Square | side × side | 11 × 11 = 121 |
| Missing side | area ÷ known side | 96 ÷ 12 = 8 |
| L-shape | split and add, or subtract | 60 − 12 = 48 |
Practice questions
- 1What is the area of a rectangle 9 cm long and 5 cm wide?
- 2What is the area of a square of side 8 m?
- 3A rectangle has an area of 36 square cm and a length of 9 cm. What is its width?
- 4A floor 12 m by 8 m is covered with 1 square metre tiles costing ₹50 each. What is the total cost?
- 5A rectangle has a perimeter of 30 cm and a length of 10 cm. What is its area?
- 6A square has a perimeter of 40 m. What is its area?
- 7From a rectangle 10 m by 6 m, a corner rectangle 4 m by 3 m is cut away to make an L-shape. What is the area of the L-shape?
- 8Among rectangles of area 24 square units with whole-number sides, which perimeter is the smallest?
Show answers (try them yourself first)
- 11) 45 square cm
- 22) 64 square m
- 33) 4 cm
- 44) ₹4800 (96 tiles)
- 55) 50 square cm (width 5)
- 66) 100 square m (side 10)
- 77) 48 square m (60 − 12)
- 88) 20 (the 4 by 6 rectangle)
Area of a triangle
A triangular flag, a triangular garden corner and the sloping end of a roof all need a way to find their area. Cutting a rectangle along a diagonal gives two triangles, which is the key idea.
The area of a triangle = half × base × height. The height is the straight (perpendicular) distance from the top corner to the base, not the length of a slanted side. Triangles with the same base and height have the same area, however far the top corner slides.
In the figure: A right triangle is half of the 8 by 5 rectangle, so its area is 20.
Rule 1: A right triangle is half a rectangle — Two equal right triangles fit together to make a rectangle. So the area of a right triangle is half the product of the two sides that meet at the right angle.
- Easy: Legs 8 cm and 6 cm: 8 × 6 = 48 and half of 48 is 24 square cm.
- Medium: Legs 10 m and 9 m: 10 × 9 = 90 and half of 90 is 45 square m.
- Exam-type: A 14 cm by 9 cm sheet cut along a diagonal: 14 × 9 = 126 and half of 126 is 63 square cm for each piece.
Rule 2: Half × base × height — For any triangle multiply the base by the height and take half. Slide the top corner along a line parallel to the base and the area does not change, because the base and height stay the same.
Area of a triangle
In the app you can move this figure with your finger.
Open in the app- Easy: Base 14 cm, height 9 cm: 14 × 9 = 126 and half is 63 square cm.
- Medium: Base 11 m, height 6 m: 11 × 6 = 66 and half is 33 square m.
- Exam-type: Area 48 square cm and base 12 cm: 48 × 2 = 96 and 96 ÷ 12 = 8 cm is the height.
Measure the height at a right angle to the base; the slanted side is never the height unless the triangle is right-angled there.
Forgetting the half, or using the slanted side as the height.
Multiply base and height first, then halve the answer. Memory clue: “Triangle = Two make a rectangle, so take Half”.
Spot the difference
| Triangle | Area | Example |
|---|---|---|
| Any triangle | ½ × base × height | ½ × 14 × 9 = 63 |
| Right triangle | ½ × the two right-angle sides | ½ × 8 × 6 = 24 |
| Missing height | 2 × area ÷ base | 2 × 48 ÷ 12 = 8 |
| Rectangle cut on a diagonal | half of the rectangle | ½ × 14 × 9 = 63 |
Practice questions
- 1A triangle has base 12 cm and height 5 cm. What is its area?
- 2A right triangle has its two short sides 6 cm and 4 cm. What is its area?
- 3A triangle has base 8 cm and height 8 cm. What is its area?
- 4A triangle has an area of 40 square cm and a base of 10 cm. What is its height?
- 5A triangle has base 16 cm and height 9 cm. What is its area?
- 6A triangular flower bed has base 18 m and height 10 m. Planting costs ₹5 per square metre. What is the total cost?
- 7A rectangular sheet 14 cm by 9 cm is cut along a diagonal. What is the area of each piece?
- 8Two triangles have the same base 12 cm, and their heights are 5 cm and 9 cm. What is their total area?
Show answers (try them yourself first)
- 11) 30 square cm
- 22) 12 square cm
- 33) 32 square cm
- 44) 8 cm (2 × 40 ÷ 10)
- 55) 72 square cm
- 66) ₹450 (area 90 square m)
- 77) 63 square cm
- 88) 84 square cm (30 + 54)