Class 6 Mathematics – NCERT (Ganita Prakash)
Chapter 6: PERIMETER AND AREA
Study Material cum Worksheet
Study Notes & Key Properties
1. Perimeter
The perimeter of a closed plane figure is the total distance covered along its boundary when you go around it once.
The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once. For a polygon, i.e., a closed plane figure made up of line segments, the perimeter is simply the sum of the lengths of its all sides, i.e., the total distance along its outer boundary. The perimeter of a polygon = the sum of the lengths of its all sidesPerimeter of a polygon = Sum of the lengths of all its sides.
Perimeter of a Rectangle = 2 × (Length + Breadth)
Perimeter of a Square = 4 × Side
Perimeter of a Triangle = Sum of the lengths of its three sides.
Perimeter of an Equilateral Triangle = 3 × Side
2. Area
The area of a closed figure is the amount of region enclosed by it.
Area is measured in square units (e.g., sq. cm, sq. m).
Area of a Rectangle = Length × Breadth
Area of a Square = Side × Side
Area of a Triangle: The area of a triangle is half the area of a rectangle with the
Regular Polygons: Closed figures with all sides and all angles equal (e.g., equilateral triangle, square). The perimeter of a regular polygon is the
3. Key Relationships
Two figures can have the same area but different perimeters.
Two figures can have the same perimeter but different areas.
Estimating Area: The area of irregular shapes can be estimated by placing them on a grid or squared paper and counting the squares.
Question Bank
Chapter 6: PERIMETER AND AREA Study Material cum Worksheet
1. Perimeter
The _______________of a closed plane figure is the total distance covered along its boundary when you go around it once.
Perimeter of a polygon = Sum of the __________ of all its sides.
Perimeter of a Rectangle =_______________
Perimeter of a Square = __________________
Perimeter of a Triangle = Sum of the lengths of its ____________sides.
Perimeter of an Equilateral Triangle = _________
The _______________of a closed plane figure is the total distance covered along its boundary when you go around it once.
Perimeter of a polygon = Sum of the __________ of all its sides.
Perimeter of a Rectangle =_______________
Perimeter of a Square = __________________
Perimeter of a Triangle = Sum of the lengths of its ____________sides.
Perimeter of an Equilateral Triangle = _________
2. Area
The _________ of a closed figure is the amount of region enclosed by it.
Area is measured in _____________ units (e.g., sq. cm, sq. m).
Area of a Rectangle = ___________
Area of a Square = ____________
Area of a Triangle: The area of a triangle is half the area of a rectangle with the same base and height. (Area =_______________)
Regular Polygons: Closed figures with all sides and all angles equal (e.g., equilateral triangle, square). The perimeter of a regular polygon is the number of __________ multiplied by the length of one side.
3. Key Relationships
Two figures can have the same area but different perimeters.
Two figures can have the same perimeter but different areas.
Estimating Area: The area of irregular shapes can be estimated by placing them on a grid or squared paper and counting the squares.
Two figures can have the same area but different perimeters.
Two figures can have the same perimeter but different areas.
Estimating Area: The area of irregular shapes can be estimated by placing them on a grid or squared paper and counting the squares.
Question Bank
Multiple Choice Questions (20 Questions)
What is the perimeter of a square with a side length of 5 cm?
a) 10 cm b) 15 cm c) 20 cm d) 25 cm (Competency: Logical Reasoning)
The area of a rectangle is 48 sq. m. If its length is 12 m, what is its breadth?
a) 4 m b) 6 m c) 8 m d) 10 m (Competency: Problem Solving)
If the perimeter of a rectangle is 30 cm and its length is 10 cm, what is its breadth?
a) 5 cm b) 10 cm c) 15 cm d) 20 cm (Competency: Logical Reasoning)
A farmer has a rectangular field of length 150 m and breadth 100 m. The total length of rope needed to fence it with 3 rounds is:
a) 250 m b) 500 m c) 1500 m d) 3000 m (Competency: Analytical Thinking)
The area of a square is 81 sq. cm. What is the length of one of its sides?
a) 9 cm b) 8 cm c) 40.5 cm d) 18 cm (Competency: Problem Solving)
The perimeter of an equilateral triangle is 21 cm. The length of each side is:
a) 3 cm b) 6 cm c) 7 cm d) 10.5 cm (Competency: Logical Reasoning)
A piece of string is 40 cm long. What will be the length of each side if it is used to form a regular pentagon?
a) 5 cm b) 8 cm c) 10 cm d) 20 cm (Competency: Problem Solving)
Four square flower beds of side 2 m are dug at the corners of a garden. What is the total area occupied by these flower beds?
a) 4 sq. m b) 8 sq. m c) 16 sq. m d) 32 sq. m (Competency: Analytical Thinking)
The cost of fencing a rectangular park at ₹20 per meter is ₹2000. If the length of the park is 30 m, its breadth is:
a) 10 m b) 20 m c) 30 m d) 40 m (Competency: Problem Solving)
If a rectangle and a square have the same perimeter, which of the following is always true?
a) They have the same area. b) The rectangle has a larger area.
c) The square has a larger area. d) Their areas cannot be compared.
(Competency: Analytical Thinking)
The area of a triangle is 24 sq. cm. If its base is 8 cm, what is its height?
a) 3 cm b) 6 cm c) 12 cm d) 16 cm (Competency: Problem Solving)
A figure is made up of two rectangles. This method of finding the total area is called:
a) Addition b) Subtraction c) Splitting d) Multiplication (Competency: Spatial Understanding)
A rectangular plot has an area of 300 sq. m. If its breadth is 15 m, the cost of fencing it at ₹25 per meter is:
a) ₹1250 b) ₹2500 c) ₹3750 d) ₹5000 (Competency: Analytical Thinking)
Which shape has a larger area if both have the same perimeter?
a) A rectangle with length 8 cm and breadth 2 cm.
b) A square with side 5 cm.
c) A rectangle with length 6 cm and breadth 4 cm.
d) A rectangle with length 7 cm and breadth 3 cm.
(Competency: Logical Reasoning)
A square is folded into two equal rectangles. Which statement is true?
a) The area of each rectangle is half the area of the square.
b) The perimeter of the square is the sum of the perimeters of the rectangles.
c) The area of the square doubles.
d) The perimeter of each rectangle is the same as the square's perimeter.
(Competency: Analytical Thinking)
To find the area of an irregular shape on a grid, we count the squares. A square that is more than half-filled is counted as:
a) 0 sq. units b) ½ sq. units c) 1 sq. unit d) 2 sq. units
(Competency: Spatial Understanding)
The perimeter of a regular hexagon with a side of 4 cm is:
a) 16 cm b) 20 cm c) 24 cm d) 28 cm (Competency: Logical Reasoning)
The length of a rectangle is twice its breadth. If its perimeter is 36 cm, its area is:
a) 54 sq. cm b) 72 sq. cm c) 81 sq. cm d) 108 sq. cm (Competency: Problem Solving)
A triangle and a parallelogram are on the same base and between the same parallels. The area of the triangle is:
a) Equal to the area of the parallelogram.
b) Half the area of the parallelogram.
c) Twice the area of the parallelogram.
d) One-fourth the area of the parallelogram.
(Competency: Analytical Thinking)
A figure has a perimeter of 36 units. If a new square unit is attached to it, the perimeter can:
a) Only increase b) Only decrease c) Increase, decrease, or remain the same d) Only remain the same (Competency: Creativity & Spatial Understanding)
What is the perimeter of a square with a side length of 5 cm?
a) 10 cm b) 15 cm c) 20 cm d) 25 cm (Competency: Logical Reasoning)
The area of a rectangle is 48 sq. m. If its length is 12 m, what is its breadth?
a) 4 m b) 6 m c) 8 m d) 10 m (Competency: Problem Solving)
If the perimeter of a rectangle is 30 cm and its length is 10 cm, what is its breadth?
a) 5 cm b) 10 cm c) 15 cm d) 20 cm (Competency: Logical Reasoning)
A farmer has a rectangular field of length 150 m and breadth 100 m. The total length of rope needed to fence it with 3 rounds is:
a) 250 m b) 500 m c) 1500 m d) 3000 m (Competency: Analytical Thinking)
The area of a square is 81 sq. cm. What is the length of one of its sides?
a) 9 cm b) 8 cm c) 40.5 cm d) 18 cm (Competency: Problem Solving)
The perimeter of an equilateral triangle is 21 cm. The length of each side is:
a) 3 cm b) 6 cm c) 7 cm d) 10.5 cm (Competency: Logical Reasoning)
A piece of string is 40 cm long. What will be the length of each side if it is used to form a regular pentagon?
a) 5 cm b) 8 cm c) 10 cm d) 20 cm (Competency: Problem Solving)
Four square flower beds of side 2 m are dug at the corners of a garden. What is the total area occupied by these flower beds?
a) 4 sq. m b) 8 sq. m c) 16 sq. m d) 32 sq. m (Competency: Analytical Thinking)
The cost of fencing a rectangular park at ₹20 per meter is ₹2000. If the length of the park is 30 m, its breadth is:
a) 10 m b) 20 m c) 30 m d) 40 m (Competency: Problem Solving)
If a rectangle and a square have the same perimeter, which of the following is always true?
a) They have the same area. b) The rectangle has a larger area.
c) The square has a larger area. d) Their areas cannot be compared.
(Competency: Analytical Thinking)
The area of a triangle is 24 sq. cm. If its base is 8 cm, what is its height?
a) 3 cm b) 6 cm c) 12 cm d) 16 cm (Competency: Problem Solving)
A figure is made up of two rectangles. This method of finding the total area is called:
a) Addition b) Subtraction c) Splitting d) Multiplication (Competency: Spatial Understanding)
A rectangular plot has an area of 300 sq. m. If its breadth is 15 m, the cost of fencing it at ₹25 per meter is:
a) ₹1250 b) ₹2500 c) ₹3750 d) ₹5000 (Competency: Analytical Thinking)
Which shape has a larger area if both have the same perimeter?
a) A rectangle with length 8 cm and breadth 2 cm.
b) A square with side 5 cm.
c) A rectangle with length 6 cm and breadth 4 cm.
d) A rectangle with length 7 cm and breadth 3 cm.
(Competency: Logical Reasoning)
A square is folded into two equal rectangles. Which statement is true?
a) The area of each rectangle is half the area of the square.
b) The perimeter of the square is the sum of the perimeters of the rectangles.
c) The area of the square doubles.
d) The perimeter of each rectangle is the same as the square's perimeter.
(Competency: Analytical Thinking)
To find the area of an irregular shape on a grid, we count the squares. A square that is more than half-filled is counted as:
a) 0 sq. units b) ½ sq. units c) 1 sq. unit d) 2 sq. units
(Competency: Spatial Understanding)
The perimeter of a regular hexagon with a side of 4 cm is:
a) 16 cm b) 20 cm c) 24 cm d) 28 cm (Competency: Logical Reasoning)
The length of a rectangle is twice its breadth. If its perimeter is 36 cm, its area is:
a) 54 sq. cm b) 72 sq. cm c) 81 sq. cm d) 108 sq. cm (Competency: Problem Solving)
A triangle and a parallelogram are on the same base and between the same parallels. The area of the triangle is:
a) Equal to the area of the parallelogram.
b) Half the area of the parallelogram.
c) Twice the area of the parallelogram.
d) One-fourth the area of the parallelogram.
(Competency: Analytical Thinking)
A figure has a perimeter of 36 units. If a new square unit is attached to it, the perimeter can:
a) Only increase b) Only decrease c) Increase, decrease, or remain the same d) Only remain the same (Competency: Creativity & Spatial Understanding)
Assertion & Reasoning Questions (20 Questions)
Directions: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
a) Both A and R are true, and R is the correct explanation of A.
b) Both A and R are true, but R is not the correct explanation of A.
c) A is true but R is false.
d) A is false but R is true.
Assertion (A): The perimeter of a rectangle with length 7 cm and breadth 3 cm is 20 cm.
Reason (R): Perimeter of a rectangle is 2 × (Length + Breadth). (Competency: Logical Reasoning)Assertion (A): A square with a side of 6 cm has a larger area than a rectangle with length 8 cm and breadth 4 cm.
Reason (R): For a given perimeter, a square has the maximum area.(Competency: Analytical Thinking)Assertion (A): The area of a triangle is always half the area of a rectangle.
Reason (R): A diagonal of a rectangle divides it into two triangles of equal area. (Competency: Spatial Understanding)Assertion (A): A regular pentagon with a side of 5 cm has a perimeter of 25 cm.
Reason (R): The perimeter of any regular polygon is the product of the number of sides and the length of one side. (Competency: Logical Reasoning)Assertion (A): Two figures with the same area must always have the same perimeter.
Reason (R): Area and perimeter are independent properties of a figure. (Competency: Analytical Thinking)Assertion (A): It is easier to measure area using square grids than using circular grids.
Reason (R): Squares tessellate without gaps, allowing for accurate measurement.
(Competency: Spatial Understanding)Assertion (A): If the side of a square is doubled, its perimeter is also doubled.
Reason (R): The perimeter of a square is directly proportional to the length of its side.
(Competency: Logical Reasoning)Assertion (A): The area of a rectangle is 40 sq. cm. If its length is 10 cm, then its breadth must be 4 cm.
Reason (R): Area of a rectangle = Length × Breadth.(Competency: Problem Solving)Assertion (A): A square is a regular polygon, but a rectangle is not.
Reason (R): A regular polygon must have all sides and all angles equal.
(Competency: Logical Reasoning)Assertion (A): When a square piece of paper is cut along its diagonal, the perimeter of each resulting triangle is greater than half the perimeter of the square.
Reason (R): The diagonal of a square is longer than its side. (Competency: Analytical Thinking)Assertion (A): The area of a path surrounding a garden is found by subtracting the area of the garden from the area of the garden including the path.
Reason (R): This method gives the area of the annular region between the two concentric figures.
(Competency: Spatial Understanding)Assertion (A): A figure made of 9 unit squares can have different perimeters.
Reason (R): The perimeter depends on the arrangement of the squares, not just the number.
(Competency: Creativity & Spatial Understanding)Assertion (A): The area of a parallelogram is base × height.
Reason (R): A parallelogram can be split and rearranged into a rectangle with the same base and height. (Competency: Spatial Understanding)Assertion (A): If the perimeter of a square is 28 cm, its area is 49 sq. cm.
Reason (R): The side of the square would be 7 cm. (Competency: Problem Solving)Assertion (A): The number of trees that can be planted in a field is given by (Area of field) ÷ (Area required per tree).
Reason (R): This formula gives the maximum number of non-overlapping units that can fit into a given area. (Competency: Analytical Thinking)Assertion (A): All equilateral triangles are regular polygons.
Reason (R): All isosceles triangles are regular polygons. (Competency: Logical Reasoning)Assertion (A): Estimating area by counting squares on a grid is an approximate method.
Reason (R): The edges of the irregular shape may not align perfectly with the grid lines.
(Competency: Spatial Understanding)Assertion (A): The area of a right-angled triangle is easy to find because its two sides are perpendicular.
Reason (R): The perpendicular sides can be taken as the base and the height.
(Competency: Problem Solving)Assertion (A): A rectangle of 12 sq. units area can have a perimeter of 14 units or 16 units.
Reason (R): Different dimensions (e.g., 3×4, 2×6) for the same area yield different perimeters.
(Competency: Analytical Thinking)Assertion (A): The concept of perimeter is used when putting a fence around a field.
Reason (R): The concept of area is used when painting a wall. (Competency: Real-life Application)
True/False Questions (10 Questions)
The perimeter of a rectangle is always greater than the perimeter of a square with the same area.
Area is measured in linear units like cm or m.
If you cut a rectangle into two pieces, the total area remains the same.
A figure with a larger perimeter will always have a larger area.
All squares are rectangles.
The area of a triangle formed by the diagonal of a rectangle is exactly half the area of the rectangle.
A regular hexagon with a side of 'a' units has a perimeter of 7a units.
To find the cost of tiling a floor, we need to find the perimeter of the floor.
Two different shapes can have the same perimeter and the same area.
The formula for the area of a parallelogram is the same as that for a rectangle.
(Competencies: Logical Reasoning, Analytical Thinking)
The perimeter of a rectangle is always greater than the perimeter of a square with the same area.
Area is measured in linear units like cm or m.
If you cut a rectangle into two pieces, the total area remains the same.
A figure with a larger perimeter will always have a larger area.
All squares are rectangles.
The area of a triangle formed by the diagonal of a rectangle is exactly half the area of the rectangle.
A regular hexagon with a side of 'a' units has a perimeter of 7a units.
To find the cost of tiling a floor, we need to find the perimeter of the floor.
Two different shapes can have the same perimeter and the same area.
The formula for the area of a parallelogram is the same as that for a rectangle.
(Competencies: Logical Reasoning, Analytical Thinking)
Short Answer Type I (2 Marks each - 15 Questions)
Find the perimeter of a square with a side length of 12.5 cm.
The area of a rectangular garden is 400 sq. m. If its length is 25 m, find its breadth.
A triangular park has sides of 15 m, 20 m, and 25 m. Find the total distance covered by a person who takes 2 rounds of the park.
Find the side of a square whose perimeter is 48 m.
The length and breadth of a rectangle are 10 cm and 8 cm respectively. Find the perimeter of a square having the same area as this rectangle.
A wire bent in the shape of a rectangle of sides 12 cm and 8 cm is straightened and rebent into a square. What is the length of the side of the square?
By splitting the figure below into rectangles, find its area. (Assume a grid where major divisions are 1 unit).
The perimeter of a regular pentagon is 35 cm. What is the length of each side?
If the cost of fencing a square park at ₹15 per meter is ₹1200, find the side of the square park.
A rectangle has a perimeter of 30 cm. If its length is 10 cm, what is its area?
State the formula for the area of a triangle and explain it using a diagram.
Define a regular polygon. Give two examples.
How many square tiles of side 20 cm would be needed to tile a rectangular floor of 4 m by 3 m?
The area of a square plot is 1600 sq. m. Find the length of its side.
Explain why the area of a rectangle is given by the product of its length and breadth.
(Competencies: Problem Solving, Logical Reasoning, Spatial Understanding)
Find the perimeter of a square with a side length of 12.5 cm.
The area of a rectangular garden is 400 sq. m. If its length is 25 m, find its breadth.
A triangular park has sides of 15 m, 20 m, and 25 m. Find the total distance covered by a person who takes 2 rounds of the park.
Find the side of a square whose perimeter is 48 m.
The length and breadth of a rectangle are 10 cm and 8 cm respectively. Find the perimeter of a square having the same area as this rectangle.
A wire bent in the shape of a rectangle of sides 12 cm and 8 cm is straightened and rebent into a square. What is the length of the side of the square?
By splitting the figure below into rectangles, find its area. (Assume a grid where major divisions are 1 unit).
The perimeter of a regular pentagon is 35 cm. What is the length of each side?
If the cost of fencing a square park at ₹15 per meter is ₹1200, find the side of the square park.
A rectangle has a perimeter of 30 cm. If its length is 10 cm, what is its area?
State the formula for the area of a triangle and explain it using a diagram.
Define a regular polygon. Give two examples.
How many square tiles of side 20 cm would be needed to tile a rectangular floor of 4 m by 3 m?
The area of a square plot is 1600 sq. m. Find the length of its side.
Explain why the area of a rectangle is given by the product of its length and breadth.
(Competencies: Problem Solving, Logical Reasoning, Spatial Understanding)
Short Answer Type II (3 Marks each - 10 Questions)
The length of a rectangular field is twice its breadth. If the perimeter of the field is 150 m, find its length and breadth. Also, calculate its area.
The floor is 6 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. Find the area of the floor that is not carpeted.
A piece of string is 60 cm long. What will be the length of each side if the string is used to form: a) A square? b) A regular hexagon?
Four square flower beds each side 1.5 m are dug on the four corners of a piece of land 10 m long and 8 m wide. What is the area of the remaining part of the land?
Find the perimeter of the following figure:
Draw any two shapes with an area of 12 square units but with different perimeters. (Use graph paper/grid lines).
A verandah 2 m wide is constructed all around a room of dimensions 8 m × 5 m. Find the area of the verandah.
The area of a rectangular sheet of paper is 120 sq. cm. If its length is 15 cm, what is its perimeter?
A farmer has a rectangular field of length 180 m and breadth 110 m. He wants to fence it with 4 rounds of rope. What is the total length of rope he will need? If the rope costs ₹5 per meter, what will be the total cost?
Explain with a diagram how a triangle has an area equal to half of a rectangle on the same base and between the same parallels.
(Competencies: Problem Solving, Analytical Thinking, Creativity)
The length of a rectangular field is twice its breadth. If the perimeter of the field is 150 m, find its length and breadth. Also, calculate its area.
The floor is 6 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. Find the area of the floor that is not carpeted.
A piece of string is 60 cm long. What will be the length of each side if the string is used to form: a) A square? b) A regular hexagon?
Four square flower beds each side 1.5 m are dug on the four corners of a piece of land 10 m long and 8 m wide. What is the area of the remaining part of the land?
Find the perimeter of the following figure:
Draw any two shapes with an area of 12 square units but with different perimeters. (Use graph paper/grid lines).
A verandah 2 m wide is constructed all around a room of dimensions 8 m × 5 m. Find the area of the verandah.
The area of a rectangular sheet of paper is 120 sq. cm. If its length is 15 cm, what is its perimeter?
A farmer has a rectangular field of length 180 m and breadth 110 m. He wants to fence it with 4 rounds of rope. What is the total length of rope he will need? If the rope costs ₹5 per meter, what will be the total cost?
Explain with a diagram how a triangle has an area equal to half of a rectangle on the same base and between the same parallels.
(Competencies: Problem Solving, Analytical Thinking, Creativity)
Long Answer Type (5 Marks each - 10 Questions)
(a) Find the area of a square whose perimeter is 64 cm.
(b) A rectangle has the same area as the square. If the length of the rectangle is 32 cm, find its breadth and perimeter.
(c) Compare the perimeters of the square and the rectangle.
A rectangular park is 85 m long and 60 m wide. A path 5 m wide is to be built outside the park. Find the area of the path. Also, find the cost of cementing it at the rate of ₹250 per 10 sq. m.
(a) Define perimeter and area.
(b) The sides of a triangle are in the ratio 3:4:5. If its perimeter is 60 cm, find its area. (Hint: Check if it's a right-angled triangle).
A design has been drawn on a wall, as shown below. The unshaded parts are to be painted blue. Calculate the area to be painted.
(a) A wire is in the shape of a rectangle of length 12 cm and breadth 8 cm. It is rebent into a square. Find the side of the square.
(b) If the same wire was rebent into a regular hexagon, what would be the length of each side?
(c) Which shape, the square or the hexagon, encloses more area?
Using the tangram pieces (Shapes A-G):
a) How many times bigger is Shape D compared to Shape C?
b) What is the area of the big square formed by all seven pieces in terms of the area of Shape C?
c) Are the perimeters of the square and a rectangle formed from these 7 pieces different or the same? Explain.
Draw a rectangle ABCD of length 10 cm and breadth 6 cm. Draw its diagonal AC.
a) Identify the two triangles formed.
b) Prove that the area of triangle ABC is half the area of the rectangle.
c) What is the perimeter of triangle ABC if AC = 11.6 cm?
A room is 12 m long and 9 m broad. It has a door of size 2 m × 1.5 m and two windows of size 1.5 m × 1 m. Find the cost of whitewashing the walls at ₹5 per sq. m. (Note: Height of the room is 3 m).
(a) Using 9 unit squares, what is the smallest possible perimeter you can achieve? Draw the arrangement.
(b) What is the largest possible perimeter? Draw the arrangement.
(c) How does the arrangement of squares affect the perimeter?
Study the floor plan of Charan's house (from the textbook).
a) Find the missing dimensions for the Small Bedroom, Utility, and Hall.
b) Calculate the total area of the house.
c) If the cost of construction is ₹2000 per sq. ft, what is the total cost for the built-up area (all rooms except Garden and Parking)?
(Competencies: Problem Solving, Analytical Thinking, Spatial Understanding, Creativity)
(a) Find the area of a square whose perimeter is 64 cm.
(b) A rectangle has the same area as the square. If the length of the rectangle is 32 cm, find its breadth and perimeter.
(c) Compare the perimeters of the square and the rectangle.
A rectangular park is 85 m long and 60 m wide. A path 5 m wide is to be built outside the park. Find the area of the path. Also, find the cost of cementing it at the rate of ₹250 per 10 sq. m.
(a) Define perimeter and area.
(b) The sides of a triangle are in the ratio 3:4:5. If its perimeter is 60 cm, find its area. (Hint: Check if it's a right-angled triangle).
A design has been drawn on a wall, as shown below. The unshaded parts are to be painted blue. Calculate the area to be painted.
(a) A wire is in the shape of a rectangle of length 12 cm and breadth 8 cm. It is rebent into a square. Find the side of the square.
(b) If the same wire was rebent into a regular hexagon, what would be the length of each side?
(c) Which shape, the square or the hexagon, encloses more area?
Using the tangram pieces (Shapes A-G):
a) How many times bigger is Shape D compared to Shape C?
b) What is the area of the big square formed by all seven pieces in terms of the area of Shape C?
c) Are the perimeters of the square and a rectangle formed from these 7 pieces different or the same? Explain.
Draw a rectangle ABCD of length 10 cm and breadth 6 cm. Draw its diagonal AC.
a) Identify the two triangles formed.
b) Prove that the area of triangle ABC is half the area of the rectangle.
c) What is the perimeter of triangle ABC if AC = 11.6 cm?
A room is 12 m long and 9 m broad. It has a door of size 2 m × 1.5 m and two windows of size 1.5 m × 1 m. Find the cost of whitewashing the walls at ₹5 per sq. m. (Note: Height of the room is 3 m).
(a) Using 9 unit squares, what is the smallest possible perimeter you can achieve? Draw the arrangement.
(b) What is the largest possible perimeter? Draw the arrangement.
(c) How does the arrangement of squares affect the perimeter?
Study the floor plan of Charan's house (from the textbook).
a) Find the missing dimensions for the Small Bedroom, Utility, and Hall.
b) Calculate the total area of the house.
c) If the cost of construction is ₹2000 per sq. ft, what is the total cost for the built-up area (all rooms except Garden and Parking)?
(Competencies: Problem Solving, Analytical Thinking, Spatial Understanding, Creativity)
Case-Based Questions (4 Sub-questions each - 5 Cases)
Case 1: The Running Track
Akshi and Toshi are running along different rectangular tracks. Akshi's outer track is 70 m by 40 m. Toshi's inner track is 60 m by 30 m. Akshi completes 5 rounds, and Toshi completes 7 rounds.
What is the perimeter of Akshi's track?
What is the total distance covered by Toshi?
Who ran a longer distance and by how much?
If they both start at the same time and run at the same speed, who will finish their total distance first?
1. Find out the total distance Akshi has covered in 5 rounds.
2. Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance?
3. Think and mark the positions as directed—
a. Mark ‘A’ at the point where Akshi will be after she ran 250 m.
b. Mark ‘B’ at the point where Akshi will be after she ran 500 m.
c. Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘C’.
d.Mark ‘X’ at the point where Toshi will be after she ran 250 m.
e. Mark ‘Y’ at the point where Toshi will be after she ran 500 m. 133 Reprint 2025-26 Ganita Prakash | Grade 6
f. Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘Z’
(Competency: Analytical Thinking)
Case 2: The Tangram Puzzle
The tangram is a puzzle consisting of seven flat shapes.
Which two shapes have the same area?
How many times bigger is Shape D compared to Shape C?
What is the area of the big square in terms of the area of Shape C?
If the perimeter of Shape C is 10 cm, can we find the perimeter of the big square? Why or why not?
Akshi and Toshi are running along different rectangular tracks. Akshi's outer track is 70 m by 40 m. Toshi's inner track is 60 m by 30 m. Akshi completes 5 rounds, and Toshi completes 7 rounds.
What is the perimeter of Akshi's track?
What is the total distance covered by Toshi?
Who ran a longer distance and by how much?
If they both start at the same time and run at the same speed, who will finish their total distance first?
1. Find out the total distance Akshi has covered in 5 rounds.
(Competency: Analytical Thinking)
The tangram is a puzzle consisting of seven flat shapes.
Which two shapes have the same area?
How many times bigger is Shape D compared to Shape C?
What is the area of the big square in terms of the area of Shape C?
If the perimeter of Shape C is 10 cm, can we find the perimeter of the big square? Why or why not?
Figure it Out
1. Explore and figure out how many pieces have the same area.
2. How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?
3. Which shape has more area: Shape D or F? Give reasons for your answer.
4. Which shape has more area: Shape F or G? Give reasons for your answer.
5. What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big? Hint: In the tangram pieces, by placing the shapes over each other, we can find out that Shapes A and B have the same area, Shapes C and E have the same area. You would have also figured out that Shape D can be exactly covered using Shapes C and E, which means Shape D has twice the area of Shape C or shape E, etc.
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?
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.
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. (Competency: Spatial Understanding & Logical Reasoning)
Case 3: Fencing a Field
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 perimeter of the field?
What is the total length of rope required?
If the rope costs ₹8 per meter, what is the total cost of fencing?
If he uses only 2 rounds of rope, how much money will he save?
(Competency: Problem Solving)
Case 4: Area Maze- In each figure, find the missing value of either the length of a side or the area of a region.
Refer to the area maze puzzles from the textbook .
For figure (a), find the missing length if the total area is 28 sq. cm.
For figure (b), find the missing area.
What is the concept used to solve these puzzles?
Create a simple area maze puzzle of your own.
(Competency: Logical Reasoning & Creativity)
Case 5: Tiling a Floor
A floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. The remaining floor is to be tiled with tiles of size 20 cm × 20 cm.
What is the area of the floor that is not carpeted?
What is the area of one tile?
How many tiles are needed to cover the uncarpeted area?
If 10% extra tiles are required for wastage, what is the final number of tiles to be purchased?
(Competency: Analytical Thinking & Real-life Application)
Case 6: Split and rejoin
A rectangular paper chit of dimension 6 cm × 4 cm is cut as shown into two equal pieces. These two pieces are joined in different ways.
For example, the arrangement a. has a perimeter of 28 cm. Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below
2. Arrange the two pieces to form a figure with a perimeter of 22 cm (Competency: Analytical Thinking & Real-life Application)
From Textbook Questions
Figure it Out
1.Find the missing terms:
a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?.
b. Perimeter of a square = 20 cm; side of a length = ?.
c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?.
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?
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.
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?
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, b. A triangle with all sides of equal length, and c. A hexagon (a six sided closed figure) with sides of equal length?
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?
Figure it Out
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.
2. The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden. 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.
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?
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.
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?
7. Drawarectangleofsize12units×8units.Drawanotherrectangle inside it, without touching the outer rectangle that occupies exactly half the area.
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 12 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.
9. Below is the house plan of Charan. It is in a rectangular plot. Some of the measurements are given. a. Find the missing measurements. b. Find out the area of his house. find out the missing dimensions and area of Sharan’s home. Some of the measurements are given. a. Find the missing measurements. b. Find out the area of his house. What are the dimensions of all the different rooms in Sharan’s house? Compare the areas and perimeters of Sharan’s house and Charan’s house.
10. Using 9 unit squares, solve the following. 1. What is the smallest perimeter possible? 2. What is the largest perimeter possible? 3. Make a figure with a perimeter of 18 units. 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
11. Find the areas of the figures below by dividing them into rectangles and triangles.
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 perimeter of the field?
What is the total length of rope required?
If the rope costs ₹8 per meter, what is the total cost of fencing?
If he uses only 2 rounds of rope, how much money will he save?
(Competency: Problem Solving)
Refer to the area maze puzzles from the textbook .
For figure (a), find the missing length if the total area is 28 sq. cm.
For figure (b), find the missing area.
What is the concept used to solve these puzzles?
Create a simple area maze puzzle of your own.
(Competency: Logical Reasoning & Creativity)
A floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid on the floor. The remaining floor is to be tiled with tiles of size 20 cm × 20 cm.
What is the area of the floor that is not carpeted?
What is the area of one tile?
How many tiles are needed to cover the uncarpeted area?
If 10% extra tiles are required for wastage, what is the final number of tiles to be purchased?
(Competency: Analytical Thinking & Real-life Application)
A rectangular paper chit of dimension 6 cm × 4 cm is cut as shown into two equal pieces. These two pieces are joined in different ways.
For example, the arrangement a. has a perimeter of 28 cm. Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below
9. Below is the house plan of Charan. It is in a rectangular plot. Some of the measurements are given. a. Find the missing measurements. b. Find out the area of his house. find out the missing dimensions and area of Sharan’s home. Some of the measurements are given. a. Find the missing measurements. b. Find out the area of his house. What are the dimensions of all the different rooms in Sharan’s house? Compare the areas and perimeters of Sharan’s house and Charan’s house.
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