Mensuration in 6th-grade math deals with measuring the areas and perimeters of various shapes. Here’s a summary of the key concepts:
1. Units of Measurement:
- We use standard units like centimeters (cm), meters (m), and millimeters (mm) to measure length.
2. Area:
- Area represents the amount of space a flat surface occupies.
- We measure area in square units (cm², m², etc.).
- Formulas are used to calculate the area of different shapes:
- Square: Area = side × side
- Rectangle: Area = length × breadth
3. Perimeter:
- Perimeter is the total length of all sides of a closed figure.
- We measure perimeter in linear units (cm, m, etc.).
- Formulas are used to calculate the perimeter of different shapes:
- Square: Perimeter = 4 × side
- Rectangle: Perimeter = 2 × (length + breadth)
4. Applications:
- Mensuration helps solve real-life problems like finding the amount of cloth needed for a dress (area) or the length of fencing required for a garden (perimeter).
Here are some additional points to consider:
- Understanding Shapes: It’s important for students to recognize basic shapes like squares, rectangles, and triangles.
- Visualization: Drawing diagrams can help students visualize shapes and understand the concepts better.
- Practice: Solving practice problems helps students solidify their understanding of formulas and apply them to various scenarios.
By grasping these concepts, students develop a foundation for understanding geometry and measurement in higher grades.
Exercise 10.1
1. Find the perimeter of each of the following figures:
Ans :
(a) The required perimeter is: 4 cm + 2 cm + 1 cm + 5 cm = 12 cm
(b) The required perimeter is: 40 cm + 35 cm + 23 cm + 35 cm = 133 cm or 1.33 m
(c) The required perimeter is: 15 cm + 15 cm + 15 cm + 15 cm = 15 cm x 4 = 60 cm
(d) The required perimeter is: 4 cm + 4 cm + 4 cm + 4 cm + 4 cm – 4 cm x 5 = 20 cm
(e) The required perimeter is: 4 cm + 0.5 cm + 2.5 cm + 2.5 cm + 0.5 cm + 4 cm + 1 cm = 15 cm
(f) The required perimeter is: 4 cm + 1 cm + 3 cm + 2 cm + 3 cm + 4 cm + 1 cm + 3 cm + 2 cm + 3 cm + 4 cm + 1 cm + 3 cm + 2 cm + 3 cm + 4 cm + 1 cm + 3 cm + 2 cm + 3 cm = 52 cm
2. The lid of a rectangular box of sides 40 cm by 10 cm is sealed all round with tape. What is the length of the tape required?
Ans :
- Identify the sides: The lid is a rectangle with a length of 40 cm and a breadth of 10 cm (given in the problem).
- Perimeter formula for rectangle: Perimeter of a rectangle = 2 (Length + Breadth)
- Calculation:
- Perimeter = 2 (40 cm + 10 cm)
- Perimeter = 2 x 50 cm
- Perimeter = 100 cm
Therefore, 100 cm of tape is required to seal the lid of the rectangular box all around.
3. A table-top measures 2 m 25 cm by 1 m 50 cm. What is the perimeter of the table-top?
Ans :
- Convert centimeters to meters for consistency:
- Length = 2 m 25 cm = 2 m + (25 cm / 100 cm/m) = 2.25 m
- Breadth = 1 m 50 cm = 1 m + (50 cm / 100 cm/m) = 1.50 m
- Perimeter formula for rectangle: Perimeter of a rectangle = 2 (Length + Breadth)
- Calculation:
- Perimeter = 2 (2.25 m + 1.50 m)
- Perimeter = 2 x 3.75 m
- Perimeter = 7.50 m
Therefore, the perimeter of the table-top is 7.50 meters.
4. What is the length of the wooden strip required to frame a photograph of length and breadth 32 cm and 21 cm respectively?
Ans :
- Identify the sides: The photograph is a rectangle with a length of 32 cm and a breadth of 21 cm.
- Perimeter formula for rectangle: Perimeter of a rectangle = 2 (Length + Breadth)
- Calculation:
- Perimeter = 2 (32 cm + 21 cm)
- Perimeter = 2 x 53 cm
- Perimeter = 106 cm
Therefore, 106 cm of wooden strip is required to frame the photograph.
5. A rectangular piece of land measures 0.7 km by 0.5 km. Each side is to be fenced with 4 rows of wires. What is the length of the wire needed?
Ans :
- Convert kilometers to meters for perimeter calculation:
- Length (m) = 0.7 km * 1000 m/km = 700 m
- Breadth (m) = 0.5 km * 1000 m/km = 500 m
- Calculate the perimeter of the land: The perimeter represents the total length of the fence required. Perimeter = 2 (Length + Breadth) Perimeter = 2 (700 m + 500 m) Perimeter = 2 * 1200 m Perimeter = 2400 m
- Account for four rows of wires: Since each side needs to be fenced with 4 rows of wires, we multiply the perimeter by 4. Total wire length = Perimeter × Number of wire rows Total wire length = 2400 m * 4
- Calculate the total length of wire needed: Total wire length = 9600 meters
Therefore, 9600 meters of wire are needed to fence the rectangular piece of land with 4 rows of wires on each side.
6. Find the perimeter of each of the following shapes:
(a) A triangle of sides 3 cm, 4 cm and 5 cm.
(b) An equilateral triangle of side 9 cm.
(c) An isosceles triangle with equal sides 8 cm each and third side 6 cm.
Ans :
(a) Triangle with sides 3 cm, 4 cm and 5 cm:
Perimeter = Side 1 + Side 2 + Side 3 Perimeter = 3 cm + 4 cm + 5 cm Perimeter = 12 cm
(b) Equilateral Triangle with side 9 cm:
An equilateral triangle has all three sides equal.
Perimeter = Side 1 + Side 2 + Side 3 (all sides are 9 cm) Perimeter = 9 cm + 9 cm + 9 cm Perimeter = 27 cm
(c) Isosceles Triangle with equal sides 8 cm and third side 6 cm:
An isosceles triangle has two sides with the same length.
Perimeter = Side 1 + Side 2 + Side 3 Perimeter = 8 cm + 8 cm + 6 cm Perimeter = 22 cm
Therefore:
- The perimeter of the triangle in (a) is 12 cm.
- The perimeter of the equilateral triangle in (b) is 27 cm.
- The perimeter of the isosceles triangle in (c) is 22 cm.
7. Find the perimeter of a triangle with sides measuring 10 cm, 14 cm and 15 cm.
Ans : To find the perimeter of a triangle, we simply add the lengths of all its sides. Here’s how to find the perimeter of the triangle with sides 10 cm, 14 cm, and 15 cm:
Perimeter = Side 1 + Side 2 + Side 3
Perimeter = 10 cm + 14 cm + 15 cm
Perimeter = 39 cm
Therefore, the perimeter of the triangle is 39cm
8. Find the perimeter of a regular hexagon with each side measuring 8 m.
Ans :
- Identify the number of sides: A hexagon has 6 sides.
- Perimeter formula for regular shapes: Perimeter of a regular polygon = Number of sides × Side length
- Calculation:
- Perimeter = 6 sides × 8 meters/side
- Perimeter = 48 meters
Therefore, the perimeter of the regular hexagon is 48 meters.
9. Find the side of the square whose perimeter is 20 m.
Ans :
- Perimeter Formula for Square: Perimeter of a square = 4 × Side length
- Given Information: We know the perimeter of the square is 20 meters.
- Solve for Side Length:
- Since the perimeter is 20 meters, we can set up the equation: 20 m = 4 × Side length
- To find the side length, we need to isolate it. Divide both sides of the equation by 4: (20 m) / 4 = (4 × Side length) / 4
- Side Length:
- Simplifying the equation, we get: 5 m = Side length
Therefore, the side of the square is 5 meters.
10. The perimeter of a regular pentagon is 100 cm. How long is its each side?
Ans :
- Perimeter = Number of sides × Side length Perimeter = 5 × s
- Given Information: We know the perimeter of the pentagon is 100 cm.
- Solve for Side Length (s):
- Set up the equation based on the given information: 100 cm = 5 × s
- To find the side length (s), isolate it by dividing both sides by 5: (100 cm) / 5 = (5 × s) / 5
- Side Length (s):
- Simplify the equation: 20 cm = s
Therefore, each side of the regular pentagon is 20 centimeters long.
11. A piece of string is 30 cm long. What will be the length of each side if the string is used to form:
(a) a square?
(b) an equilateral triangle?
(c) a regular hexagon?
Ans :
(a) Square:
- In a square, all sides are equal in length.
- Perimeter of the square = Length of the string (given as 30 cm)
- Let the side of the square be ‘s’ cm.
- Perimeter = 4 × s (formula for square’s perimeter)
- Substitute the known value: 30 cm = 4 × s
Solving for s (side of the square):
- Divide both sides by 4: 30 cm / 4 = (4 × s) / 4
- Simplify: s = 7.5 cm
Therefore, each side of the square would be 7.5 cm long.
(b) Equilateral Triangle:
- In an equilateral triangle, all sides are equal in length.
- Perimeter of the equilateral triangle = Length of the string (30 cm)
- Let the side of the triangle be ‘s’ cm.
- Perimeter = 3 × s (formula for equilateral triangle’s perimeter)
- Substitute the known value: 30 cm = 3 × s
Solving for s (side of the equilateral triangle):
- Divide both sides by 3: 30 cm / 3 = (3 × s) / 3
- Simplify: s = 10 cm
Therefore, each side of the equilateral triangle would be 10 cm long.
(c) Regular Hexagon:
- In a regular hexagon, all sides are equal in length.
- Perimeter of the regular hexagon = Length of the string (30 cm)
- Let the side of the hexagon be ‘s’ cm.
- Perimeter = 6 × s (formula for regular hexagon’s perimeter)
- Substitute the known value: 30 cm = 6 × s
Solving for s (side of the regular hexagon):
- Divide both sides by 6: 30 cm / 6 = (6 × s) / 6
- Simplify: s = 5 cm
Therefore, each side of the regular hexagon would be 5 cm long.
In summary:
- Square: Each side = 7.5 cm
- Equilateral Triangle: Each side = 10 cm
- Regular Hexagon: Each side = 5 cm
12. Two sides of a triangle are 12 cm and 14 cm. The perimeter of the triangle is 36 cm. What is its third side?
Ans :
- Identify the given information:
- Two sides of the triangle: 12 cm and 14 cm
- Perimeter of the triangle: 36 cm
- Set up the equation:
- Let the third side of the triangle be “x” cm.
- Perimeter = Side 1 + Side 2 + Side 3
- Substitute the known values: 36 cm = 12 cm + 14 cm + x cm
- Solve for x (third side):
- Combine the known side lengths: 36 cm = 26 cm + x cm
- To isolate x, subtract the combined known side lengths from both sides: 36 cm – 26 cm = (26 cm + x cm) – 26 cm
- Simplify: 10 cm = x cm
Therefore, the third side of the triangle is 10 centimeters long.
13. Find the cost of fencing a square park of side 250 m at the rate of? 20 per metre.
Ans :
- Calculate the perimeter of the square park:
- A square has all sides equal.
- Perimeter of a square = 4 × Side length (given as 250 m)
- Perimeter = 4 × 250 m = 1000 m
- Identify the rate per meter:
- The cost per meter of fencing is 20 per metre (given).
- Calculate the total cost of fencing:
- Total cost = Perimeter × Rate per meter
- Total cost = 1000 m × 20 per metre = 20000
Therefore, the cost of fencing the square park is ₹20,000.
14. Find the cost of fencing a rectangular park of length 175 m and breadth 125 m at the rate of ₹12 per metre.
Ans :
- Calculate the perimeter of the park:
- A rectangle’s perimeter is the total length of all its sides added together.
- Perimeter = 2 (Length + Breadth)
- Given: Length (l) = 175 m and Breadth (b) = 125 m
- Perimeter = 2 (175 m + 125 m)
- Perimeter = 2 x 300 m
- Perimeter = 600 m
- Identify the rate per meter:
- The cost per meter of fencing is ₹12 (given).
- Calculate the total cost of fencing:
- Total cost = Perimeter × Rate per meter
- Total cost = 600 m × ₹12 per meter
- Total cost = ₹7200
Therefore, the cost of fencing the rectangular park is ₹7200.
15 . Sweety runs around a square park of side 75 m. Bulbul runs around a rectangular park with length 60 m and breadth 45 m. Who covers less distance?
Ans :
- Calculate the distance covered by Sweety (square park):
- Perimeter of a square = 4 × Side length
- Side length of Sweety’s park (square) = 75 m (given)
- Perimeter of Sweety’s park = 4 × 75 m = 300 m
- Calculate the distance covered by Bulbul (rectangular park):
- Perimeter of a rectangle = 2 (Length + Breadth)
- Length of Bulbul’s park = 60 m (given)
- Breadth of Bulbul’s park = 45 m (given)
- Perimeter of Bulbul’s park = 2 (60 m + 45 m) = 2 × 105 m = 210 m
- Comparison:
- Sweety’s park perimeter (distance covered) = 300 m
- Bulbul’s park perimeter (distance covered) = 210 m
Conclusion:
Bulbul covers less distance (210 m) compared to Sweety (300 m).
16. What is the perimeter of each of the following figures? What do you infer from the answers?
Ans :
(a) The perimeter of the square is calculated as: 25 cm + 25 cm + 25 cm + 25 cm = 4 x 25 cm = 100 cm
(b) The perimeter of the rectangle is calculated as: 30 cm + 20 cm + 30 cm + 20 cm = 2 x (30 cm + 20 cm) = 2 x 50 cm = 100 cm
(c) The perimeter of the rectangle is calculated as: 40 cm + 10 cm + 40 cm + 10 cm = 2 x (40 cm + 10 cm) = 2 x 50 cm = 100 cm
(d) The perimeter of the triangle is the sum of all sides: 30 cm + 30 cm + 40 cm = 100 cm
From the above answers, we conclude that different shapes can have equal perimeters.
17. Avneet buys 9 square paving slabs, each with a side of 1/7 m. He lays them in the form of a square.
(a) What is the perimeter of his arrangement [Fig. (i)]?
(b) Shari does not like his arrangement. She gets him to lay them out like a cross. What is the perimeter of her arrangement [Fig. (ii)]?
(c) Which has greater perimeter?
(d) Avneet wonders, if there is a way of getting an even greater perimeter. Can you find a way of doing this? (The paving slabs must meet along complete edges, i.e., they can not be broken).
Ans :
(a) The arrangement is in the form of a square of side
(b) Perimeter of cross-arrangement
(c) Since 10 m > 6 m
∴ Cross-arrangement has greater perimeter.
(d) Total number of tiles = 9
∴ We have the following arrangement
The above arrangement will also have the greater perimeter.
Exercise 10.2
Find the areas of the following figures by counting square:
Ans :
(a) Number of full squares = 9
Area of 1 square = 1 sq unit
∴ Area of 9 squares = 9 x 1 sq unit = 9 sq units
So, the area of the portion covered by 9 squares = 9 sq units
(b) Number of full squares = 5
∴ Area of the figure = 5 x 1 sq unit = 5 sq units
(c) Number of full squares = 2
Number of half squares = 4
∴ Area of the covered figure = 2 x 1 + 4 x 1/2 = 2 + 2 = 4 sq units
(d) Number of full squares = 8
∴ Area of the covered portion of the figure = 8 x 1 sq unit = 8 sq units
(e) Number of full squares = 10
Area covered by the figure = 10 x 1 sq unit = 10 sq units
(f) Number of full squares = 2 Number of half squares = 4 ∴ Area of the covered figure = (2 x 1 + 4 x 1/2) = (2 + 2) sq units = 4 sq units
(g) Number of full squares = 4
Number of half squares = 4
∴ Area of the covered figure = (4 x 1 + 4 x 1/2) = (4 + 2) sq units = 6 sq units
(h) Number of full squares = 5
∴ Area of the covered figure = 5 x 1 sq unit = 5 sq units
(i) Number of full squares = 9
∴ Area of the covered figure = 9 x 1 sq unit = 9 sq units
(j) Number of full squares = 2
Number of half squares = 4
∴ Area of the covered figure = (2 x 1 + 4 x 1/2) sq units = (2 + 2) sq units = 4 sq units
(k) Number of full squares = 4 Number of half squares = 2 ∴ Area of the covered figure = (4 x 1 + 2 x 1/2) sq units = (4 + 1) sq units = 5 sq units
(l) Number of full squares = 4
Number of squares more than half = 3
Number of half squares = 2
∴ Area of the covered figure = (4 x 1 + 3 x 1 + 2 x 1/2) sq units = (4 + 3 + 1) sq units = 8 sq units
(m) Number of full squares = 6
Number of more than half squares = 8
Area of the covered figure = (6 x 1 + 8 x 1) sq units = (6 + 8) sq units = 14 sq units
(n) Number of full squares = 9
Number of more than half squares = 9
∴ Area of the covered figure = (9 x 1 + 9 x 1) sq units = (9 + 9) sq units = 18 sq units
Exercise 10.3
1. Find the areas of the rectangles whose sides are:
(a) 3 cm and 4 cm
(b) 12 m and 21 m
(c) 2 km and 3 km
(d) 2 m and 70 cm
Ans :
(a) Sides: 3 cm and 4 cm Area = 3 cm × 4 cm = 12 cm²
(b) Sides: 12 m and 21 m Area = 12 m × 21 m = 252 m²
(c) Sides: 2 km and 3 km (convert kilometers to meters for consistency) 1 km = 1000 m Area = 2 km × 3 km = (2 × 1000 m) × (3 × 1000 m) = 6,000,000 m²
(d) Sides: 2 m and 70 cm (convert centimeters to meters for consistency) 100 cm = 1 m Area = 2 m × (70 cm / 100 cm/m) = 2 m × 0.7 m = 1.4 m²
2. Find the areas of the squares whose sides are:
(a) 10 cm
(b) 14 cm
(c) 5 m
Ans :
Area of a square = Side length × Side length
(a) Side: 10 cm
Area = 10 cm × 10 cm = 100 cm²
(b) Side: 14 cm
Area = 14 cm × 14 cm = 196 cm²
(c) Side: 5 m
Area = 5 m × 5 m = 25 m²
3. The length and breadth of three rectangles are as given below:
(a) 9 m and 6 m
(b) 17 m and 3 m
(c) 4 m and 14 m
Ans :
Area = Length × Breadth
Finding the Area for Each Rectangle:
Now, let’s find the area of each rectangle using the given information:
(a) Length = 9 m, Breadth = 6 m
Area = 9 m × 6 m = 54 square meters (m²)
(b) Length = 17 m, Breadth = 3 m
Area = 17 m × 3 m = 51 square meters (m²)
(c) Length = 4 m, Breadth = 14 m
Area = 4 m × 14 m = 56 square meters (m²)
4. The area of a rectangular garden 50 m long is 300 sq m. Find the width of the garden.
Ans :
- Length of the garden (l) = 50 meters (given)
- Area of the garden (A) = 300 square meters (given)
- Understand the relationship between area and dimensions of a rectangle: The area of a rectangle is the product of its length and width. We can express this with the formula: A = l × b (where A is area, l is length, and b is width)
- Solve for the width (b):
- We need to isolate the width (b) in the formula.
- Since we know the length (l) and the area (A), we can rearrange the formula to solve for b: b = A / l
- Calculate the width (b):
- Substitute the known values: b = 300 m² / 50 m b = 6 meters
Therefore, the width of the rectangular garden is 6 meters.
5. 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?
Ans :
- Calculate the area of the plot:
- Area of a rectangle = Length × Breadth
- Length (l) = 500 m (given)
- Breadth (b) = 200 m (given)
- Area = 500 m × 200 m = 100,000 square meters (sq m)
- Understand the rate per unit area:
- The rate is given as ₹8 per hundred sq m. This means the cost applies to every 100 sq m of the plot.
- Calculate the total cost:
- We need to find out how many times the cost of ₹8 applies to the entire area (100,000 sq m).
- Divide the total area by the area covered by the given rate: Number of times the rate applies = 100,000 sq m / 100 sq m/unit = 1000 units
- Multiply the number of units by the cost per unit to find the total cost: Total cost = Number of units × Cost per unit Total cost = 1000 units × ₹8/unit = ₹8000
Therefore, the cost of tiling the rectangular plot of land is ₹8000.
6. A table-top measures 2 m by 1 m 50 cm. What is its area in square metres?
Ans :
- We know the width is 1 meter and 50 centimeters (1 m 50 cm).
- 1 centimeter (cm) is equal to 0.01 meters (m).
- Width in meters = 1 m + (50 cm * 0.01 m/cm)
- Width in meters = 1 m + 0.5 m
- Width in meters = 1.5 m
Calculate the area:
- Area of a rectangle = Length × Breadth
- Length (l) = 2 meters (given)
- Breadth (b) = 1.5 meters (calculated)
- Area = 2 m × 1.5 m = 3 square meters (m²)
Therefore, the area of the table-top is 3 square meters.
7. A room is 4 m long and 3 m 50 cm wide. How many square metres of carpet is needed to cover the floor of the room?
Ans :
- The width is given as 3 meters and 50 centimeters (3 m 50 cm).
- We know 1 centimeter (cm) is equal to 0.01 meters (m).
- Width in meters = 3 m + (50 cm * 0.01 m/cm)
- Width in meters = 3 m + 0.5 m
- Width in meters = 3.5 meters
Calculate the room’s area:
- Area of a rectangle = Length × Breadth
- Length (l) = 4 meters (given)
- Breadth (b) = 3.5 meters (calculated)
- Area = 4 m × 3.5 m = 14 square meters (m²)
Therefore, you will need 14 square meters of carpet to cover the floor of the room.
8. A floor is 5 m long and 4 m wide. A square carpet of sides 3 m is laid on the floor. Find the area of the floor that is not carpeted.
Ans :
- Calculate the area of the floor:
- Area of a rectangle = Length × Breadth
- Length of the floor (l) = 5 meters (given)
- Breadth of the floor (b) = 4 meters (given)
- Area of the floor = 5 m × 4 m = 20 square meters (m²)
- Calculate the area of the square carpet:
- Area of a square = Side length × Side length
- Side length of the carpet (s) = 3 meters (given)
- Area of the carpet = 3 m × 3 m = 9 square meters (m²)
- Area of the non-carpeted floor:
- The non-carpeted area is the difference between the total floor area and the carpeted area.
- Area of non-carpeted floor = Area of the floor – Area of the carpet
- Area of non-carpeted floor = 20 m² – 9 m² = 11 square meters (m²)
Therefore, the area of the floor that is not carpeted is 11 square meters.
9. Five square flower beds each of side 1 m are dug on a piece of land 5 m long and 4 m wide. What is the area of the remaining part of the land?
Ans :
- Calculate the area of one flower bed:
- Area of a square = Side length × Side length
- Side length of a flower bed (s) = 1 meter (given)
- Area of one flower bed = 1 m × 1 m = 1 square meter (m²)
- Calculate the total area of the five flower beds:
- Number of flower beds = 5 (given)
- Total area of flower beds = Number of flower beds × Area of one flower bed
- Total area of flower beds = 5 × 1 m² = 5 square meters (m²)
- Calculate the area of the whole piece of land:
- Area of the land = Length × Breadth
- Length of the land (l) = 5 meters (given)
- Breadth of the land (b) = 4 meters (given)
- Area of the land = 5 m × 4 m = 20 square meters (m²)
- Calculate the area of the remaining land:
- The remaining land is the difference between the total land area and the total area occupied by the flower beds.
- Area of remaining land = Area of the land – Total area of flower beds
- Area of remaining land = 20 m² – 5 m² = 15 square meters (m²)
Therefore, the area of the remaining part of the land is 15 square meters.
10. By splitting the following figures into rectangles, find their areas (The measures are given in centimetres).
Ans :
Figure (a):
- Splitting into rectangles:
- The figure can be split into two rectangles: Rectangle I and Rectangle II.
- Rectangle I has dimensions 3 cm × 4 cm (from the image).
- Rectangle II has dimensions 4 cm × 3 cm (from the image).
- Calculating the area of each rectangle:
- Area of Rectangle I = 3 cm × 4 cm = 12 cm²
- Area of Rectangle II = 4 cm × 3 cm = 12 cm²
- Calculating the total area of the figure:
- The total area is the sum of the areas of both rectangles.
- Total area of figure (a) = Area of Rectangle I + Area of Rectangle II
- Total area of figure (a) = 12 cm² + 12 cm² = 24 cm²
Figure (b):
- Splitting into rectangles:
- The figure can be split into three rectangles: Rectangle I, Rectangle II, and Rectangle III (all with the same dimensions).
- Each rectangle has dimensions 3 cm × 1 cm (from the image).
- Calculating the area of each rectangle:
- Area of each rectangle = 3 cm × 1 cm = 3 cm²
- Calculating the total area of the figure:
- The total area is the sum of the areas of all three rectangles.
- Total area of figure (b) = 3 cm² + 3 cm² + 3 cm² = 9 cm²
11. Split the following shapes into rectangles and find their areas (The measures are given in centimetres).
Ans :
(a) By splitting the given figure into rectangles I and II, we get:
Area of rectangle I: = 12 cm x 2 cm = 24 sq cm
Area of rectangle II: = 8 cm x 2 cm = 16 sq cm
∴ The total area of the whole figure is: 24 sq cm + 16 sq cm = 40 sq cm.
(b) By splitting the given figure into rectangles I, II, and III, we get:
Area of rectangle I: = 7 cm x 7 cm = 49 sq cm
Area of rectangle II: = 21 cm x 7 cm = 147 sq cm
Area of rectangle III: = 7 cm x 7 cm = 49 sq cm
∴ The total area of the whole figure is: 49 sq cm + 147 sq cm + 49 sq cm = 245 sq cm.
12. How many tiles whose length and breadth are 12 cm and 5 cm respectively will be needed to fit in a rectangular region whose length and breadth are respectively:
(a) 100 cm and 144 cm
(b) 70 cm and 36 cm
Ans :
1. Calculate the area of one tile:
- Area of a tile = Length × Breadth
- Tile length = 12 cm (given)
- Tile breadth = 5 cm (given)
- Area of one tile = 12 cm × 5 cm = 60 cm²
2. Find the area of the rectangular region for each case:
(a) Length = 100 cm, Breadth = 144 cm:
- Area of the region = Length × Breadth
- Area = 100 cm × 144 cm = 14400 cm²
(b) Length = 70 cm, Breadth = 36 cm:
- Area of the region = Length × Breadth
- Area = 70 cm × 36 cm = 2520 cm²
3. Calculate the number of tiles required for each case:
(a) Number of tiles for region (a) = Area of region (a) / Area of one tile
- Number of tiles = 14400 cm² / 60 cm² = 240 tiles
(b) Number of tiles for region (b) = Area of region (b) / Area of one tile
- Number of tiles = 2520 cm² / 60 cm² = 42 tiles
