Class 6 › Mathematics › Ganita Prakash Perimeter and Area Class 6 Mathematics · Ganita Prakash · Chapter 6
Practice all → Summary Questions & answers Practise this chapter → This chapter studies perimeter and area of plane figures. The perimeter of any closed figure is the distance covered along its boundary in one round; for a polygon it is the sum of all side lengths. The perimeter of a rectangle is \(2 \times (\text{length} + \text{breadth})\), of a square is \(4 \times \text{side}\), and of a triangle is the sum of its three sides; a regular polygon\u2019s perimeter is the number of sides times the length of a side. Area is the amount of region enclosed by a closed figure, measured in square units. The area of a rectangle is length \(\times\) breadth and the area of a square is side \(\times\) side. The chapter shows how to estimate areas of irregular shapes using squared or graph paper, counting full squares and half squares. A key result is that the area of a triangle is half the area of the rectangle that contains it, so the area of a right triangle is half of base \(\times\) height. Through tangram puzzles, house plans and area mazes, students see that two figures can have the same area but different perimeters, or the same perimeter but different areas.
Perimeter
The total distance along the boundary of a closed figure, i.e. the sum of all its side lengths.
Area
The amount of region enclosed by a closed figure, measured in square units.
Perimeter of a rectangle
Twice the sum of length and breadth: \(2 \times (l + b)\).
Perimeter of a square
Four times the length of a side: \(4 \times \text{side}\).
Area of a rectangle
Length multiplied by breadth: \(l \times b\).
Square unit
The unit of area, equal to the area of a \(1 \times 1\) square, such as a square centimetre or square metre.
Q1 What is the perimeter of a closed figure? Answer Q2 What is the formula for the perimeter of a rectangle? Answer Q3 What is the perimeter of a square of side \(s\)? Answer Q5 What is the area of a rectangle? Answer Q6 What is the area of a square of side \(s\)? Answer Q7 How is the area of a right triangle related to a rectangle? Answer Q8 How can you estimate the area of an irregular shape? Answer Q9 Can two figures with the same area have different perimeters? Answer Q10 Find the width of a rectangular garden 25 m long with area 300 sq m. Answer Q1 1. Find the missing terms: a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?. Answer Q2 1. Find the missing terms: b. Perimeter of a square = 20 cm; side of a length = ?. Answer Q3 1. Find the missing terms: c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?. Answer Q4 2. A rectangle having sidelengths 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? Answer Q5 3. 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, respectively. Answer Q6 4. 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? Answer Q7 5. 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, Answer Q8 5. A piece of string is 36 cm long. What will be the length of each side, if it is used to form: b. A triangle with all sides of equal length, and Answer Q9 5. A piece of string is 36 cm long. What will be the length of each side, if it is used to form: c. A hexagon (a six sided closed figure) with sides of equal length? Answer Q10 6. A farmer has a rectangular field having length 230 m and breadth 160 m. He wants to fence it with 3 rounds of rope as shown. What is the total length of rope needed? Answer Q1 1. Find out the total distance Akshi has covered in 5 rounds. Answer Q2 2. Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance? Answer Q3 3. Think and mark the positions as directed— a. Mark ‘A’ at the point where Akshi will be after she ran 250 m. Answer Q4 3. Think and mark the positions as directed— b. Mark ‘B’ at the point where Akshi will be after she ran 500 m. Answer Q5 3. Think and mark the positions as directed— c. Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘C’. Answer Q6 3. Think and mark the positions as directed— d. Mark ‘X’ at the point where Toshi will be after she ran 250 m. Answer Q7 3. Think and mark the positions as directed— e. Mark ‘Y’ at the point where Toshi will be after she ran 500 m. Answer Q8 3. Think and mark the positions as directed— f. Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘Z’. Answer Q1 In races, usually there is a common finish line for all the runners. Here are two square running tracks with the inner track of 100 m each side and outer track of 150 m each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks. If the total race is of 350 m, then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for 350 m. Mark the starting points of the runner on the inner track as ‘A’ and the runner on the outer track as ‘B’. Answer Q1 Take a rough sheet of paper or a sheet of newspaper. Make a few random shapes by cutting the paper in different ways. Estimate the total length of the boundaries of each shape then use a scale or measuring tape to measure and verify the perimeter for each shape. Answer Q1 Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think? Answer Q1 Write the perimeters of the figures below in terms of straight and diagonal units. Answer Q1 Find various objects from your surroundings that have regular shapes and find their perimeters. Also, generalise your understanding for the perimeter of other regular polygons. Answer Q1 Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. b. Answer Q2 Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. c. Answer Q3 Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. d. Answer Q1 Arrange the two pieces to form a figure with a perimeter of 22 cm. Answer Q1 1. The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden? Answer Q2 2. 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? Answer Q3 3. 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 in this grove? Answer Q4 4. By splitting the following figures into rectangles, find their areas (all measures are given in metres). a. Answer Q5 4. By splitting the following figures into rectangles, find their areas (all measures are given in metres). b. Answer Q1 1. Explore and figure out how many pieces have the same area. Answer Q2 2. How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E? Answer Q3 3. Which shape has more area: Shape D or F? Give reasons for your answer. Answer Q4 4. Which shape has more area: Shape F or G? Give reasons for your answer. Answer Q5 5. What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big? Answer Q6 6. Can you now figure out the area of the big square formed with all seven pieces in terms of the area of Shape C? Answer Q7 7. Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle in terms of the area of Shape C now? Give reasons for your answer. Answer Q8 8. Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same? Give an explanation for your answer. Answer Q1 Find the area of the following figures. Answer Q1 Why is area generally measured using squares? Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region. As you can see, circles can’t be packed tightly without gaps in between. So, it is difficult to get an accurate measurement of area using circles as units. Here, the same rectangle is packed in two different ways with circles—the first one has 42 circles and the second one has 44 circles. Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area. Answer Q2 1. Find the area (in square metres) of the floor outside of the corridor. Answer Q3 2. Find the area (in square metres) occupied by your school playground. Answer Q1 Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. a. Which rectangle has the greatest perimeter? Answer Q2 Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. b. Which rectangle has the least perimeter? Answer Q3 Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. c. If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer. Answer Q1 Draw a rectangle on a piece of paper and draw one of its diagonals. Cut the rectangle along that diagonal and get two triangles. Check! whether the two triangles overlap each other exactly. Do they have the same area? Answer Q2 Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Why? Can you see some relationship between the blue rectangle and the yellow triangle and their areas? Write the relationship here. Answer Q3 1. Find the area of blue triangle BAD. Answer Q4 2. Find the area of red triangle ABE. Answer Q1 1. Find the areas of the figures below by dividing them into rectangles and triangles. a. Answer Q2 1. Find the areas of the figures below by dividing them into rectangles and triangles. b. Answer Q3 1. Find the areas of the figures below by dividing them into rectangles and triangles. c. Answer Q4 1. Find the areas of the figures below by dividing them into rectangles and triangles. d. Answer Q5 1. Find the areas of the figures below by dividing them into rectangles and triangles. e. Answer Q1 Using 9 unit squares, solve the following. 1. What is the smallest perimeter possible? Answer Q2 Using 9 unit squares, solve the following. 2. What is the largest perimeter possible? Answer Q3 Using 9 unit squares, solve the following. 3. Make a figure with a perimeter of 18 units. Answer Q4 Using 9 unit squares, solve the following. 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning? Answer Q5 We have a figure below having perimeter 24 units. Without calculating all over again, observe, think and find out what will be the change in the perimeter if a new square is attached as shown on the right. Answer Q6 Experiment placing this new square at different places and think what the change in perimeter will be. Can you place the square so that the perimeter: a) increases; b) decreases; c) stays the same? Answer Q1 Below is the house plan of Charan. It is in a rectangular plot. Look at the plan. What do you notice? Some of the measurements are given. a. Find the missing measurements. Answer Q2 Below is the house plan of Charan. It is in a rectangular plot. Look at the plan. What do you notice? b. Find out the area of his house. Answer Q1 Now, find out the missing dimensions and area of Sharan’s home. Below is the plan: Some of the measurements are given. a. Find the missing measurements. Answer Q2 Now, find out the missing dimensions and area of Sharan’s home. Below is the plan: Some of the measurements are given. b. Find out the area of his house. Answer Q1 In each figure, find the missing value of either the length of a side or the area of a region. a. Answer Q2 In each figure, find the missing value of either the length of a side or the area of a region. b. Answer Q3 In each figure, find the missing value of either the length of a side or the area of a region. c. Answer Q4 In each figure, find the missing value of either the length of a side or the area of a region. d. Answer Q1 1. Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: 5 m × 10 m and 2 m × 7 m. Answer Q2 2. The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden. Answer Q3 3. The floor of a room is 5 m long and 4 m wide. A square carpet whose sides are 3 m in length is laid on the floor. Find the area that is not carpeted. Answer Q4 4. 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 laying down a lawn? Answer Q5 5. 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 satisfying the given conditions. Answer Q6 6. On a page in your book, draw a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border? Answer Q7 7. 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. Answer Q8 8. A square piece of paper is folded in half. The square is then cut into two rectangles along the fold. Regardless of the size of the square, one of the following statements is always true. Which statement is true here? a. The area of each rectangle is larger than the area of the square. b. The perimeter of the square is greater than the perimeters of both the rectangles added together. c. The perimeters of both the rectangles added together is always 1 1/2 times the perimeter of the square. d. The area of the square is always three times as large as the areas of both rectangles added together. Answer
The perimeter of a closed figure is
A the region enclosed by it B the number of its corners C the distance along its boundary in one round D half its longest side
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