6ನೇ ತರಗತಿ ಗಣಿತ (ಗಣಿತ ಪ್ರಕಾಶ) ಭಾಗ-2ರ 6ನೇ ಅಧ್ಯಾಯ “ಸುತ್ತಳತೆ ಮತ್ತು ವಿಸ್ತೀರ್ಣ” (Perimeter and Area) ಪಾಠದ ಸರಳ ಇಂಗ್ಲಿಷ್ ಟಿಪ್ಪಣಿಗಳು ಇಲ್ಲಿವೆ. ಆಯತ, ಚೌಕ ಮತ್ತು ತ್ರಿಭುಜದ ಸುತ್ತಳತೆ, ನಿಯಮಿತ ಬಹುಭುಜಾಕೃತಿಗಳು, ವಿಸ್ತೀರ್ಣ, ಟ್ಯಾಂಗ್ರಾಮ್, ಚೌಕ ಹಾಳೆಯಲ್ಲಿ ವಿಸ್ತೀರ್ಣ ಅಂದಾಜು ಮತ್ತು ತ್ರಿಭುಜದ ವಿಸ್ತೀರ್ಣವನ್ನು ಪಠ್ಯಪುಸ್ತಕದ ಚಿತ್ರಗಳೊಂದಿಗೆ ವಿವರಿಸಲಾಗಿದೆ. ಎಲ್ಲಾ “Figure it Out” ಪ್ರಶ್ನೆಗಳಿಗೆ ಹಂತ ಹಂತದ ಉತ್ತರಗಳು, ಆಟಗಳು, ಉಚಿತ PDF ಗಳು ಮತ್ತು ಶಾಲೆಯ ಹೆಸರಿನಲ್ಲಿ ಬಳಸಬಹುದಾದ ಪ್ರಶ್ನೆಪತ್ರಿಕೆ (Word) ಲಭ್ಯವಿದೆ.
Perimeter and Area
The full lesson in simple words with textbook maps and pictures, animations, games and quizzes, followed by answers to every question and activity.
Questions and answers: all Figure it Out questions, puzzles and activities.
Tangram pieces
Area of a triangle on grid paperThe Lesson
How far is it around a shape, and how much space does it cover? Let us find out.
6.1 Perimeter
The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once.
For a polygon (a closed plane figure made up of line segments), the perimeter is simply the sum of the lengths of all its sides.
Perimeter of a rectangle
Rectangle ABCD has length 12 cm and breadth 8 cm. Opposite sides of a rectangle are always equal, so AB = CD and AD = BC.
Perimeter = AB + BC + CD + DA = AB + BC + AB + BC = 2 × (AB + BC) = 2 × (12 cm + 8 cm) = 2 × 20 cm = 40 cm.
Perimeter of a rectangle = 2 × (length + breadth). It is twice the sum of its length and breadth.
Perimeter of a square
Debojeet wants to put coloured tape all around a square photo frame of side 1 m. Tape needed = perimeter = 1 m + 1 m + 1 m + 1 m = 4 m.
All four sides of a square are equal, so we can multiply: 4 × 1 m = 4 m.
Perimeter of a square = 4 × length of a side.
Perimeter of a triangle
A triangle has sides 4 cm, 5 cm and 7 cm. Perimeter = 4 cm + 5 cm + 7 cm = 16 cm.
Perimeter of a triangle = sum of the lengths of its three sides.
Examples
- Akshi’s lace: a rectangular tablecloth is 3 m long and 2 m wide. Lace = perimeter = 2 × (3 m + 2 m) = 2 × 5 m = 10 m.
- Usha’s run: a square park has side 75 m. One round = 4 × 75 m = 300 m. Three rounds = 3 × 300 m = 900 m.
✦ Matha Pachchi! Running tracks
Akshi runs on the outer rectangular track (70 m × 40 m) and Toshi on the inner track (60 m × 30 m). One round of Akshi’s track = 2 × (70 + 40) m = 220 m.
Akshi and Toshi on the rectangular tracks
Square tracks with a common finishing line✦ Straight and diagonal units
On dot paper, some lines join dots side by side (straight units) and some join dots corner to corner (diagonal units). A diagonal unit is longer than a straight unit.
A triangle made of 6 straight units and 3 diagonal units has perimeter 6s + 3d units. So its perimeter is more than 9 units.
Akshi and Toshi’s triangle on dot paper
Figures on dot paper✦ Perimeter of a regular polygon
Closed figures that have all sides and all angles equal are called regular polygons, for example the equilateral triangle, the square and the regular pentagon.
Perimeter of an equilateral triangle = AB + BC + AC = AB + AB + AB = 3 × length of a side.
Split and rejoin
A rectangular paper chit of 6 cm × 4 cm is cut into two equal pieces (each 6 cm × 2 cm). The pieces are joined in different ways. Arrangement a. has a perimeter of 28 cm.
The 6 cm × 4 cm chit cut into two pieces
Arrangement a
Arrangements b, c and d6.2 Area
The amount of region enclosed by a closed figure is called its area.
Area of a rectangle = length × width. Area of a square = side × side.
Examples
- Carpet: a floor is 5 m × 4 m = 20 sq m. A square carpet of side 3 m covers 3 m × 3 m = 9 sq m. Floor not carpeted = 20 sq m − 9 sq m = 11 sq m.
- Flower beds: land is 12 m × 10 m = 120 sq m. Four square beds of side 4 m = 4 × 16 sq m = 64 sq m. Remaining land = 120 sq m − 64 sq m = 56 sq m.
Tangram
Cut out the tangram pieces given at the end of the textbook. By placing the shapes over each other, Shapes A and B have the same area, Shapes C and E have the same area, and Shape D can be exactly covered by Shapes C and E.
TangramEstimating area with squared paper
- The area of one full small square is taken as 1 sq unit.
- Ignore portions that are less than half a square.
- If more than half of a square is in the region, count it as 1 sq unit.
- If exactly half the square is counted, take its area as ½ sq unit.
Which has a larger area?Why squares? Circles can’t be packed tightly without gaps, so it is difficult to get an accurate measurement of area using circles. The same rectangle was packed with 42 circles one way and 44 circles another way.
The same rectangle packed with 42 and 44 circles6.3 Area of a triangle
Draw a rectangle and one of its diagonals. Cut along the diagonal: the two triangles overlap exactly, so they have the same area. Each triangle is half of the rectangle.
Blue rectangle and yellow triangleIn rectangle ABCD, triangle BAD is half of the rectangle. For triangle ABE, area of ABE = area of AEF + area of BEF = half of rectangle AFED + half of rectangle BFEC = half of the area of rectangle ABCD.
Rectangle ABCD on grid paper✦ Making it ‘More’ or ‘Less’
Using 9 unit squares (area 9 sq units), we can make figures with different perimeters: a 3 × 3 square has a perimeter of 12 units, while another figure has a perimeter of 20 units.
So two figures can have the same area but different perimeters.
Two figures made with 9 unit squares
Adding a new square
Plan of Charan’s house
Plan of Sharan’s house
Area maze puzzles★ Summary
- The perimeter of a polygon is the sum of the lengths of all its sides.
- Perimeter of a rectangle = 2 × (length + width); perimeter of a square = 4 × side.
- The area of a closed figure is the measure of the region enclosed by it. Area is generally measured in square units.
- Area of a rectangle = length × width; area of a square = side × side.
- Two closed figures can have the same area with different perimeters, or the same perimeter with different areas.
- Areas can be estimated (or even found exactly) by breaking regions into unit squares, or into rectangles and triangles.
Perimeter, polygon, regular polygon, equilateral triangle, area, square unit, diagonal, tangram.
Game 1: Perimeter or Area?
Read the card and tap the answer. 8 cards!
Game 2: Quick calculation
- Perimeter of a square of side 5 cm
- Perimeter of a rectangle 7 cm × 3 cm
- Area of a rectangle 6 m × 4 m
- Area of a square of side 7 cm
- Perimeter of an equilateral triangle of side 6 cm
- Area of a triangle that is half of a 8 cm × 4 cm rectangle
Game 3: Match the formula
Questions & Answers
Figure it Out and in-text questions pages 4–21
Figure it Out (page 4)
Find the missing terms: a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ? b. Perimeter of a square = 20 cm; side = ? c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?
Page 4- a. length + breadth = 14 ÷ 2 = 7 cm, so length = 7 − 2 = 5 cm.
- b. side = 20 ÷ 4 = 5 cm.
- c. length + breadth = 12 ÷ 2 = 6 m, so breadth = 6 − 3 = 3 m.
A rectangle having side lengths 5 cm and 3 cm is made using a piece of wire. If the wire is straightened and then bent to form a square, what will be the length of a side of the square?
Page 4Length of wire = perimeter of rectangle = 2 × (5 + 3) = 16 cm. Side of square = 16 ÷ 4 = 4 cm.
Find the length of the third side of a triangle having a perimeter of 55 cm and having two sides of length 20 cm and 14 cm.
Page 4Third side = 55 − (20 + 14) = 55 − 34 = 21 cm.
What would be the cost of fencing a rectangular park whose length is 150 m and breadth is 120 m, if the fence costs ₹40 per metre?
Page 4Perimeter = 2 × (150 + 120) = 2 × 270 = 540 m. Cost = 540 × ₹40 = ₹21,600.
A piece of string is 36 cm long. What will be the length of each side if it is used to form: a. a square, b. a triangle with all sides of equal length, c. a hexagon with sides of equal length?
Page 4- a. 36 ÷ 4 = 9 cm
- b. 36 ÷ 3 = 12 cm
- c. 36 ÷ 6 = 6 cm
A farmer has a rectangular field of length 230 m and breadth 160 m. He wants to fence it with 3 rounds of rope. What is the total length of rope needed?
Page 4One round = 2 × (230 + 160) = 2 × 390 = 780 m. Three rounds = 3 × 780 = 2,340 m.
Matha Pachchi, perimeter puzzles and split and rejoin (pages 5–8)
Find out the total distance Akshi has covered in 5 rounds.
Page 5One round = 220 m. 5 rounds = 5 × 220 = 1,100 m.
Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance?
Page 5Toshi’s round = 2 × (60 + 30) = 180 m. 7 rounds = 7 × 180 = 1,260 m. Toshi ran the longer distance (1,260 m > 1,100 m).
Mark the positions: A (Akshi after 250 m), B (Akshi after 500 m), C (Akshi after 1000 m; how many full rounds?), X (Toshi after 250 m), Y (Toshi after 500 m), Z (Toshi after 1000 m; how many full rounds?).
Page 5Both start at the bottom-right corner and run along the bottom side first, then up the left side, along the top and down the right side.
- A: 250 = 220 + 30 → 1 round, then 30 m along the bottom side from the start.
- B: 500 = 2 × 220 + 60 → 2 rounds, then 60 m along the bottom side (10 m before the bottom-left corner).
- C: 1000 = 4 × 220 + 120 → 4 full rounds; then 70 m (bottom) + 40 m (left side) + 10 m along the top side.
- X: 250 = 180 + 70 → 1 round, then 60 m (bottom) + 10 m up the left side.
- Y: 500 = 2 × 180 + 140 → 2 rounds, then 60 m + 30 m + 50 m along the top side (10 m before the top-right corner).
- Z: 1000 = 5 × 180 + 100 → 5 full rounds; then 60 m + 30 m + 10 m along the top side.
Deep Dive: The inner square track is 100 m on each side and the outer one is 150 m on each side. For a 350 m race with the common finishing line at the centre of the bottom side, mark the starting points A (inner) and B (outer).
Page 6Count 350 m backwards from the finishing line.
- Inner track (A): 50 m (to the bottom corner) + 100 m + 100 m + 100 m = 350 m. So A is at the corner where the runner starts up the right side (bottom-right corner of the inner track).
- Outer track (B): 75 m (to the bottom corner) + 150 m (side) + 125 m along the top = 350 m. So B is on the top side, 25 m from the top-right corner.
Akshi says the perimeter of this triangle is 9 units. Toshi says it will be more than 9 units. What do you think?
Page 6Toshi is right. The perimeter is 6 straight units + 3 diagonal units = 6s + 3d. A diagonal unit is longer than a straight unit, so the perimeter is more than 9 units.
Write the perimeters of the figures in terms of straight and diagonal units.
Page 7- Purple figure: 8s + 2d
- Pink figure: 4s + 6d
- Green figure: 12s + 6d
- Brown figure (N): 18s + 6d
What is a similarity between a square and an equilateral triangle?
Page 7Both are regular polygons: all their sides are equal and all their angles are equal. So their perimeter = number of sides × length of a side (square: 4 × side; equilateral triangle: 3 × side).
Find objects with regular shapes, find their perimeters and generalise the perimeter of a regular polygon.
Page 8For any regular polygon, all sides are equal, so perimeter = number of sides × length of one side. (Measure objects around you and use this rule.)
Split and rejoin: find the perimeters of arrangements b, c and d. Arrange the two pieces to form a figure with a perimeter of 22 cm.
Page 8Each piece is 6 cm × 2 cm, perimeter 16 cm; two pieces = 32 cm. Where the pieces touch, that length is lost from both pieces.
- b. touching length 2 cm → 32 − 2 × 2 = 28 cm
- c. touching length 2 cm → 32 − 4 = 28 cm
- d. touching length 3 cm → 32 − 2 × 3 = 26 cm
- 22 cm: the touching length must be 5 cm (32 − 10 = 22). Place the two pieces side by side along their long sides with one piece moved 1 cm, so 5 cm of their long sides touch.
Figure it Out (page 10)
The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden?
Page 10Width = 300 ÷ 25 = 12 m.
What is the cost of tiling a rectangular plot of land 500 m long and 200 m wide at the rate of ₹8 per hundred sq m?
Page 10Area = 500 × 200 = 1,00,000 sq m = 1,000 hundreds of sq m. Cost = 1,000 × ₹8 = ₹8,000.
A rectangular coconut grove is 100 m long and 50 m wide. If each coconut tree requires 25 sq m, what is the maximum number of trees that can be planted?
Page 10Area = 100 × 50 = 5,000 sq m. Number of trees = 5,000 ÷ 25 = 200 trees.
By splitting the figures into rectangles, find their areas (all measures in metres).
Page 10
Figures for question 4 (in metres)- a. Split into strips: 3 × 3 = 9, 7 × 1 = 7, 5 × 2 = 10, 2 × 1 = 2. Area = 9 + 7 + 10 + 2 = 28 sq m.
- b. Top strip 5 × 1 = 5, two legs 1 × 2 = 2 each. Area = 5 + 2 + 2 = 9 sq m (or 5 × 3 − 3 × 2 = 9 sq m).
Tangram: Figure it Out (page 11)
Explore and figure out how many pieces have the same area.
Page 11Shapes A and B have the same area; Shapes C and E have the same area; Shapes D, F and G have the same area.
How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?
Page 11Shape D is 2 times Shape C. Shape D can be exactly covered by Shapes C and E: D = C + E.
Which shape has more area: Shape D or F? Give reasons.
Page 11Both have the same area. Each can be covered exactly by the two small triangles C and E, so each is 2 times Shape C.
Which shape has more area: Shape F or G? Give reasons.
Page 11Both have the same area (each is 2 times Shape C); they can be covered by Shapes C and E.
What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big?
Page 11Shape A = 4 times Shape C and Shape G = 2 times Shape C. So Shape A is twice as big as Shape G.
What is the area of the big square formed with all seven pieces in terms of the area of Shape C?
Page 11A + B + C + D + E + F + G = 4 + 4 + 1 + 2 + 1 + 2 + 2 = 16 times the area of Shape C.
Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle in terms of Shape C?
Page 1116 times Shape C. The same seven pieces are used without overlaps or gaps, so the area does not change.
Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same?
Page 11Different. The area is the same but the boundary changes when the pieces are rearranged; the rectangle has a longer perimeter than the square. Figures with the same area can have different perimeters.
Estimating area (pages 12–14)
Look at the two figures and guess which one has a larger area.
Page 12Trace each figure on squared paper and count the squares using the rules (full square = 1, more than half = 1, half = ½, less than half = 0). The figure with more squares has the larger area.
Find the area of the figures on dot paper (each square = 1 sq unit).
Page 12- Purple figure: 4 sq units
- Pink figure: 9 sq units
- Green figure: 10 sq units
- Brown figure (N): 11 sq units
Draw a circle of diameter 3 on a graph sheet and estimate its area by counting squares.
Page 13Counting full squares and squares that are more than half inside gives about 7 sq units.
List the points that make a square the best shape to measure area.
Page 13- Squares fit together with no gaps and no overlaps (circles leave gaps).
- Squares fill rectangles exactly in rows and columns, so we can count easily.
- All sides of a square are equal, so the unit is the same in every direction.
Find the area (in square metres) of the floor outside of the corridor.
Page 13Measure the length and width of the floor with a measuring tape and multiply: area = length × width. (Answers depend on your school.)
Find the area (in square metres) occupied by your school playground.
Page 13Measure the length and width of the playground and multiply them. If it is not a rectangle, split it into rectangles, find each area and add. (Answers depend on your school.)
Make rectangles with whole-number sides and area 24 square units. a. Which has the greatest perimeter? b. Which has the least? c. What if the area is 32 sq cm? Can you predict the shapes?
Page 13| Rectangle (24 sq units) | Perimeter |
|---|---|
| 1 × 24 | 50 units |
| 2 × 12 | 28 units |
| 3 × 8 | 22 units |
| 4 × 6 | 20 units |
- a. Greatest: 1 × 24 (50 units). b. Least: 4 × 6 (20 units).
- c. For 32 sq cm: 1 × 32 (66 cm), 2 × 16 (36 cm), 4 × 8 (24 cm). Greatest: 1 × 32; least: 4 × 8.
- Yes: the longest, thinnest rectangle has the greatest perimeter, and the rectangle closest to a square has the least perimeter.
Area of a triangle (pages 14–16)
Cut a rectangle along a diagonal. Do the two triangles have the same area? What can you infer?
Page 14Yes, the two triangles overlap exactly, so they have the same area. A diagonal divides a rectangle (or square) into two equal triangles; each triangle has half the area of the rectangle.
Is the area of the blue rectangle more or less than the area of the yellow triangle, or the same? What is the relationship?
Page 14The same. The yellow triangle is made of the two halves of the blue rectangle (cut along its diagonal), so both have equal area.
Find the area of rectangle ABCD, blue triangle BAD and red triangle ABE. What is your conclusion?
Page 15- Area of rectangle ABCD = 5 × 4 = 20 sq units.
- Area of triangle BAD = half of 20 = 10 sq units.
- Area of triangle ABE = half of rectangle AFED (12) + half of rectangle BFEC (8) = 6 + 4 = 10 sq units.
- Conclusion: a triangle drawn on the side of a rectangle with its top vertex on the opposite side has half the area of the rectangle.
Find the areas of figures a to e by dividing them into rectangles and triangles (1 square = 1 sq unit).
Page 16
Figures a to e on grid paper- a. 24 sq units
- b. 30 sq units
- c. 48 sq units
- d. 16 sq units (rectangle 4 × 5 = 20 minus the triangle cut out = 4)
- e. 12 sq units (top triangle 4 + bottom triangle 8)
Making it ‘More’ or ‘Less’ (pages 17–19)
Using 9 unit squares, what is the smallest perimeter possible?
Page 1712 units, made by a 3 × 3 square.
What is the largest perimeter possible?
Page 1720 units, for example all 9 squares in one straight line (or any shape where squares touch only one after another).
Make a figure with a perimeter of 18 units.
Page 17Example: 7 squares in a row with 2 more squares placed under the first two squares. Perimeter = 18 units.
Can you make other shaped figures for each of these perimeters, or is there only one shape?
Page 17For 12 units there is only one shape, the 3 × 3 square. For 18 units and 20 units many different shapes are possible, because the squares can be joined in many ways with the same number of touching sides.
A figure has perimeter 24 units. What happens to the perimeter when a new square is attached as shown? Can you place the square so that the perimeter increases, decreases or stays the same?
Page 17- In the place shown, the new square touches the figure on 2 sides: it adds 2 units and hides 2 units, so the perimeter stays the same (24 units).
- Increases: attach it touching only 1 side → perimeter becomes 26 units.
- Decreases: place it in a gap where it touches 3 sides → perimeter becomes 22 units.
- Stays the same: place it touching 2 sides.
Charan’s house plan (plot 35 ft × 30 ft): a. Find the missing measurements. b. Find the area of his house.
Page 18| Room | Size | Area |
|---|---|---|
| Master bedroom | 15 ft × 15 ft | 225 sq ft |
| Toilet | 5 ft × 10 ft | 50 sq ft |
| Utility | 15 ft × 3 ft | 45 sq ft |
| Kitchen | 15 ft × 12 ft | 180 sq ft |
| Small bedroom | 15 ft × 12 ft | 180 sq ft |
| Hall | 20 ft × 12 ft + 5 ft × 5 ft | 265 sq ft |
| Garden | 20 ft × 3 ft | 60 sq ft |
| Parking | 15 ft × 3 ft | 45 sq ft |
b. Area of the house = 35 ft × 30 ft = 1,050 sq ft (the sum of all parts is also 1,050 sq ft).
Sharan’s house plan (width 42 ft): find the missing dimensions and the area. Compare the areas and perimeters of Sharan’s and Charan’s houses.
Page 19| Room | Size | Area |
|---|---|---|
| Master bedroom | 12 ft × 15 ft | 180 sq ft |
| Small bedroom | 12 ft × 10 ft | 120 sq ft |
| Toilet | 5 ft × 10 ft | 50 sq ft |
| Kitchen | 18 ft × 10 ft | 180 sq ft |
| Utility | 7 ft × 10 ft | 70 sq ft |
| Hall | 23 ft × 15 ft | 345 sq ft |
| Entrance | 7 ft × 15 ft | 105 sq ft |
- Sharan’s house: 42 ft × 25 ft = 1,050 sq ft; perimeter = 2 × (42 + 25) = 134 ft.
- Charan’s house: 35 ft × 30 ft = 1,050 sq ft; perimeter = 2 × (35 + 30) = 130 ft.
- Both houses have the same area but different perimeters.
Area maze puzzles: find the missing length or area.
Page 20- a. The right column is twice as wide as the left (26 = 2 × 13). So ? = 2 × 15 = 30 sq cm.
- b. Bottom piece: 10 ÷ 2 = 5 cm long, so the tall piece is 5 − 3 = 2 cm wide and 10 ÷ 2 = 5 cm tall. The pink square is 5 − 2 = 3 cm tall and 3 cm wide: ? = 9 sq cm.
- c. Middle piece: 42 ÷ 6 = 7 cm wide. Bottom piece: 7 + 5 = 12 cm wide, so 60 ÷ 12 = 5 cm tall. Pink piece: 15 − 6 − 5 = 4 cm tall and 7 − 3 = 4 cm wide: ? = 16 sq cm.
- d. Small piece: 18 ÷ 5 = 3.6 cm tall. Big piece: 4 + 3.6 = 7.6 cm tall, so ? = 38 ÷ 7.6 = 5 cm.
Figure it Out (page 21)
Give the dimensions of a rectangle whose area is the sum of the areas of rectangles 5 m × 10 m and 2 m × 7 m.
Page 21Total area = 50 + 14 = 64 sq m. A rectangle of 8 m × 8 m (or 16 m × 4 m, or 32 m × 2 m) has this area.
The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden.
Page 21Width = 1000 ÷ 50 = 20 m.
The floor of a room is 5 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. Find the area that is not carpeted.
Page 21Floor = 5 × 4 = 20 sq m; carpet = 3 × 3 = 9 sq m. Not carpeted = 20 − 9 = 11 sq m.
Four flower beds having sides 2 m long and 1 m wide are dug at the four corners of a garden that is 15 m long and 12 m wide. How much area is now available for a lawn?
Page 21Garden = 15 × 12 = 180 sq m; beds = 4 × (2 × 1) = 8 sq m. Lawn = 180 − 8 = 172 sq m.
Shape A has an area of 18 square units and Shape B has an area of 20 square units. Shape A has a longer perimeter than Shape B. Draw two such shapes.
Page 21Example: Shape A = 18 squares in a straight line (1 × 18): area 18, perimeter 38 units. Shape B = a 4 × 5 rectangle: area 20, perimeter 18 units. Shape A has the longer perimeter.
On a page of your book, draw a rectangular border 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border?
Page 21Measure the page. Border length = page length − 2 cm; border width = page width − 3 cm. Perimeter of border = 2 × (border length + border width) = perimeter of the page − 10 cm.
Draw a rectangle of size 12 units × 8 units. Draw another rectangle inside it, without touching the outer rectangle, that occupies exactly half the area.
Page 21Outer area = 12 × 8 = 96 sq units; half = 48 sq units. Draw an inner rectangle of 8 units × 6 units (= 48 sq units) in the middle; it fits without touching (8 < 12 and 6 < 8).
A square piece of paper is folded in half and cut into two rectangles along the fold. Which statement is always true? a. Area of each rectangle is larger than the square. b. Perimeter of the square is greater than the perimeters of both rectangles together. c. The perimeters of both rectangles together is always 1½ times the perimeter of the square. d. Area of the square is three times the areas of both rectangles together.
Page 21c. is true. If the side of the square is s, its perimeter is 4s. Each rectangle is s × s/2 with perimeter 2 × (s + s/2) = 3s. Together = 6s = 1½ × 4s.
