Perimeter is the total distance around a two-dimensional shape. It is calculated by adding the lengths of all the sides of the shape.
Area is the amount of space a two-dimensional shape covers. It is measured in square units.
Key shapes and their formulas:
- Rectangle:
- Perimeter = 2(length + breadth)
- Area = length * breadth
- Square:
- Perimeter = 4 * side
- Area = side * side
- Triangle:
- Area = (1/2) * base * height
- Parallelogram:
- Area = base * height
- Circle:
- Circumference (perimeter) = 2πr (where r is the radius)
- Area = πr² (where r is the radius)
Important points:
- Units for perimeter are units of length (e.g., cm, m).
- Units for area are square units (e.g., cm², m²).
- The value of π is approximately 3.14 or 22/7.
This chapter focuses on understanding these concepts, applying formulas to calculate perimeter and area for different shapes, and solving real-world problems involving these measurements.
Exercise 9.1
1. Find the area of each of the following parallelograms:
Ans :
Area of a parallelogram = Base × Height
a) Base = 7 cm, Height = 4 cm Area = 7 cm × 4 cm = 28 cm²
b) Base = 5 cm, Height = 3 cm Area = 5 cm × 3 cm = 15 cm²
c) Base = 2.5 cm, Height = 3.5 cm Area = 2.5 cm × 3.5 cm = 8.75 cm²
d) Base = 5 cm, Height = 4.8 cm Area = 5 cm × 4.8 cm = 24 cm²
e) Base = 2 cm, Height = 4.4 cm Area = 2 cm × 4.4 cm = 8.8 cm²
2. Find the area of each of the following triangles:
Ans :
Area of a triangle
= (1/2) * base * height
Triangle (a):
- Base = 4 cm
- Height = 3 cm
Area = (1/2) * 4 cm * 3 cm = 6 cm²
Triangle (b):
- Base = 5 cm
- Height = 3.2 cm
Area = (1/2) * 5 cm * 3.2 cm = 8 cm²
Triangle (c):
- Base = 3 cm
- Height = 4 cm
Area = (1/2) * 3 cm * 4 cm
= 6 cm²
Triangle (d):
- Base = 3 cm
- Height = 2 cm
Area = (1/2) * 3 cm * 2 cm
= 3 cm²
3. Find the missing values:
| S.No. | Base | Height | Area of the parallelogram |
| (a) | 20 cm | 246 cm2 | |
| (6) | 15 cm | 154.5 cm2 | |
| (c) | 8.4 cm | 48.72 cm2 | |
| (d) | 15.6 | 16.38 cm2 |
Ans :
Area of a parallelogram = Base × Height
(a) Base = 20 cm, Height = ? (Area = 246 cm²)
- Substitute the known values: 246 cm² = 20 cm × Height
- Solve for Height: Height = 246 cm² / 20 cm = 12.3 cm
(b) Base = 15 cm, Height = 15 cm (Area = ? cm²)
- Substitute the known values: Area = 15 cm × 15 cm
- Calculate the Area: Area = 225 cm²
(c) Base = 8.4 cm, Height = ? (Area = 48.72 cm²)
- Substitute the known values: 48.72 cm² = 8.4 cm × Height
- Solve for Height: Height = 48.72 cm² / 8.4 cm ≈ 5.8 cm (rounded to two decimal places)
(d) Base = ? cm, Height = 16.38 cm (Area = 15.6 cm²)
- Substitute the known values: 15.6 cm² = Base × 16.38 cm
- Solve for Base: Base = 15.6 cm² / 16.38 cm ≈ 0.95 cm (rounded to two decimal places)
| S.No. | Base (cm) | Height (cm) | Area of the parallelogram (cm²) |
| (a) | 20 | 12.3 | 246 |
| (b) | 15 | 15 | 225 |
| (c) | 8.4 | ≈ 5.8 | 48.72 |
| (d) | ≈ 0.95 | 16.38 | 15.6 |
4. Find the missing values:
| Base | Height | Area of the triangle |
| 15 cm | — | 87 cm2 |
| — | 31.4 mm | 1256 mm2 |
| 22 cm | — | 170.5 cm2 |
Ans :
Row 1:
Base = 15 cm
Height = ?
Area = 87 cm²
Using the formula:
87 cm²
= (1/2) * 15 cm * height
height = (87 * 2) / 15
= 11.6 cm
Row 2:
Base = ?
Height = 31.4 mm
Area = 1256 mm²
Using the formula:
1256 mm²
= (1/2) * base * 31.4 mm
base = (1256 * 2) / 31.4 = 80 mm
Row 3:
Base = 22 cm
Height = ?
Area = 170.5 cm²
Using the formula:
170.5 cm² = (1/2) * 22 cm * height
height = (170.5 * 2) / 22
= 15.5 cm
| Base (cm) | Height (cm) | Area (cm²) |
| 15 | 11.6 | 87 |
| 80 | 31.4 | 1256 |
| 22 | 15.5 | 170.5 |
5. PQRS is a parallelogram. QM is the height of Q to SR and QN is the height from Q to PS. If SR = 12 cm and QM = 7.6 cm. Find:
(a) the area of the parallelogram PQRS
(b) QN, if PS = 8 cm
Ans :
(a) Area of parallelogram PQRS
- Area = base * height
Here, the base is SR = 12 cm and the corresponding height is QM = 7.6 cm.
So, Area of PQRS = 12 cm * 7.6 cm = 91.2 cm²
(b) QN, if PS = 8 cm
6. DL and BM are the heights on sides AB and AD respectively of parallelogram ABCD. If the area of the parallelogram is 1470 cm2, AB = 35 cm and AD = 49 cm, find the length of BM and DL.
Ans :
We have a parallelogram ABCD with:
- Area = 1470 cm²
- AB = 35 cm
- AD = 49 cm
- DL is the height perpendicular to AB
- BM is the height perpendicular to AD
We need to find the lengths of DL and BM.
Solution
Formula
- Area = base * height
Finding DL:
- Area = AB * DL
- 1470 cm² = 35 cm * DL
- DL = 1470 cm² / 35 cm
- DL = 42 cm
Finding BM:
- Area = AD * BM
- 1470 cm² = 49 cm * BM
- BM = 1470 cm² / 49 cm
- BM = 30 cm
Therefore, the length of DL is 42 cm and the length of BM is 30 cm.
7. ∆ABC is right angled at A. AD is perpendicular to BC. If AB = 5 cm, BC = 13 cm and AC = 12 cm, find the area of ∆ABC. Also find the length of AD.
Ans :
We have a right-angled triangle ABC with:
- AB = 5 cm
- BC = 13 cm
- AC = 12 cm
- AD is perpendicular to BC
We need to find:
- Area of triangle ABC
- Length of AD
Solution
Finding the Area of Triangle ABC
- Area = (1/2) * base * height
Here, base = AB = 5 cm and height = AC = 12 cm
So, Area of triangle ABC = (1/2) * 5 cm * 12 cm = 30 cm²
Finding the Length of AD: We know that triangle ABC and triangle ADB are similar triangles.
AB/BC = AD/AC
- 5/13 = AD/12
Cross-multiplying, we get:
- 13 * AD = 5 * 12
- AD = (5 * 12) / 13
- AD = 60/13 cm
8. ∆ABC is isosceles with AB = AC = 7.5 cm and BC = 9 cm. The height AD from A to BC, is 6 cm. Find the area of ∆ABC. What will be the height from C to AB i.e., CE?
Ans :
We have an isosceles triangle ABC with:
- AB = AC = 7.5 cm
- BC = 9 cm
- Height AD = 6 cm
We need to find:
- Area of triangle ABC
- Height CE from C to AB
Solution
Finding the Area of Triangle ABC: Since we know the base BC and the corresponding height AD, we can use the formula for the area of a triangle:
- Area = (1/2) * base * height
So, Area of triangle ABC = (1/2) * 9 cm * 6 cm = 27 cm²
Finding the Height CE: Since triangle ABC is isosceles, the area can also be calculated using AB as the base and CE as the height.
So, Area of triangle ABC = (1/2) * AB * CE
We know the area and AB, so we can find CE: 27 cm² = (1/2) * 7.5 cm * CE CE = (27 * 2) / 7.5 = 7.2 cm
Therefore, the area of triangle ABC is 27 cm² and the height CE is 7.2 cm.
Exercise 9.2
1. Find the circumference of the circles with the following radius. (Take = 22/7)
(a) 14 cm
(b) 28 mm
(c) 21 cm
Ans :
Formula:
Circumference of a circle
= 2 * π * radius
Given:
π = 22/7
Calculations:
(a) Radius = 14 cm Circumference = 2 * (22/7) * 14 cm
= 88 cm
(b) Radius = 28 mm Circumference = 2 * (22/7) * 28 mm
= 176 mm
(c) Radius = 21 cm Circumference = 2 * (22/7) * 21 cm
= 132 cm
2. Find the area of the following circles, given that (Take π =22/7)
(a) radius = 14 mm
(b) diameter = 49 m
(c) radius = 5 cm
Ans :
Formula:
Area of a circle = π * r²
where:
- π (pi) = 22/7
- r = radius of the circle
Calculations:
(a) Radius = 14 mm Area = (22/7) * (14)² mm² = (22/7) * 196 mm² = 616 mm²
(b) Diameter = 49 m First, find the radius: Radius = Diameter / 2 = 49 m / 2 = 24.5 m Area = (22/7) * (24.5)² m² = (22/7) * 600.25 m² ≈ 1886.5 m²
(c) Radius = 5 cm Area = (22/7) * (5)² cm² = (22/7) * 25 cm² ≈ 78.57 cm²
3. If the circumference of a circular sheet is 154 m, find its radius. Also find the area of the sheet. (Take π =22/7)
Ans :
4. A gardener wants to fence a circular garden of diameter 21 m. Find the length of the rope he needs to purchase, if he makes 2 rounds offence. Also find the cost of the rope, if it costs ₹ 4 per metre. (Take π =22/7)
Ans :
We are given:
- Diameter of the circular garden = 21 m
- Cost of rope per meter = ₹4
We need to find:
- Total length of rope required for 2 rounds of fencing
- Total cost of the rope
Solution
Step 1: Finding the radius Radius of the garden = Diameter / 2 = 21 m / 2 = 10.5 m
Step 2: Finding the circumference Circumference of the garden = 2 * π * radius = 2 * (22/7) * 10.5 m = 66 m
Step 3: Finding the total length of rope Since the gardener wants to make 2 rounds of fencing, the total length of rope required = 2 * 66 m = 132 m
Step 4: Finding the total cost of the rope Cost of 1 meter rope = ₹4 Cost of 132 meters rope = ₹4 * 132 = ₹528
Therefore, the gardener needs to purchase 132 meters of rope, and the total cost of the rope is ₹528.
5. From a circular sheet of radius 4 cm, a circle of radius 3 cm is removed. Find the area of the remaining sheet. (Take π = 3.14)
Ans :
Step 1:
Radius of the larger circle (R)
= 4 cm
Area of the larger circle = πR² = 3.14 * (4 cm)² = 50.24 cm²
Step 2:
Radius of the smaller circle ®
= 3 cm
Area of the smaller circle = πr² = 3.14 * (3 cm)² = 28.26 cm²
Step 3:
Area of the remaining sheet = Area of larger circle – Area of smaller circle = 50.24 cm² – 28.26 cm² = 21.98 cm²
Therefore, the area of the remaining sheet is 21.98 cm².
6. Saima wants to put a lace on the edge of a circular table cover of diameter 1.5 m. Find the length of the lace required and also find its cost if one metre of the lace costs ₹ 15. (Take π = 3.14)
Ans :
Diameter of the circular table cover
= 1.5 m
Cost of 1 meter lace = ₹15
We need to find the length of lace required and its total cost.
Solution
Step 1: Find the radius of the table cover
Radius = Diameter / 2 = 1.5 m / 2
= 0.75 m
Step 2: Find the length of the lace required
Length of the lace = Circumference of the table cover
Circumference = 2 * π * radius = 2 * 3.14 * 0.75 m = 4.71 m
Step 3: Find the total cost of the lace
Cost of 1 meter lace = ₹15
Cost of 4.71 meters lace = 4.71 * ₹15 = ₹70.65
Therefore, the length of the lace required is 4.71 meters and its total cost is ₹70.65.
7. Find the perimeter of the given figure, which is a semicircle including its diameter.
Ans :
Calculations:
Radius: Since the diameter is 10 cm, the radius is half of that:
Radius = Diameter / 2 = 10 cm / 2 = 5 cm
- Circumference of the full circle:
Circumference = 2 * π * radius = 2 * π * 5 cm
= 10π cm
- Circumference of the semicircle:
Circumference of semicircle = (Circumference of full circle) / 2 = (10π cm) / 2 = 5π cm - Perimeter of the figure:
Perimeter = Circumference of semicircle + Diameter = 5π cm + 10 cm - Result:
The perimeter of the given figure is 5π + 10 cm.
If you’d like a numerical approximation, you can substitute π with its approximate value (3.14) to get:
Perimeter ≈ 5 * 3.14 + 10 ≈ 25.7 cm
8. Find the cost of polishing a circular table-top of diameter 1.6 m, if the rate of polishing is ₹ 15 m2. (Take π = 3.14)
Ans :
Diameter of the circular table-top = 1.6 m
Rate of polishing = ₹15/m²
π = 3.14
Solution:
Step 1: Find the radius of the table-top
- Radius = Diameter / 2 = 1.6 m / 2 = 0.8 m
Step 2: Find the area of the table-top
- Area of a circle = π * radius²
- Area = 3.14 * (0.8 m)² = 3.14 * 0.64 m² = 2.0096 m²
Step 3: Find the cost of polishing
- Cost = Area * Rate of polishing
- Cost = 2.0096 m² * ₹15/m² = ₹30.144
Therefore, the cost of polishing the circular table-top is ₹30.14.
9. Shazli took a wire of length 44 cm and bent it into the shape of a circle. Find the radius of that circle. Also find its area. If the same wire is bent into the shape of a square, what will be the length of each of its sides? Which figure encloses more area, the circle or the square? (Take π =22/7)
Ans :
Circle
1. Finding the radius:
The wire is bent into a circle, so its length becomes the circumference of the circle.
Circumference = 2πr = 44 cm
So, r = 44 / (2 * 22/7) = 7 cm
2. Finding the area:
Area of a circle = πr²
Area = (22/7) * 7 * 7 = 154 cm²
Square
1. Finding the side length:
The wire is bent into a square, so its length becomes the perimeter of the square.
Perimeter of a square = 4 * side = 44 cm
So, side = 44 / 4
= 11 cm
2. Finding the area:
Area of a square = side * side
Area = 11 * 11
= 121 cm²
Comparison
Area of the circle is 154 cm²
Area of the square is 121 cm²
The circle encloses more area than the square.
10.From a circular card sheet of radius 14 cm, two circles of radius 3.5 cm and a rectangle of length 3 cm and breadth 1 cm are removed, (as shown in the given figure below). Find the area of the remaining sheet. (Take π = 22/7)
Ans :
Step 1: Find the area of the large circular sheet:
- Radius of the large circle (R) = 14 cm
- Area of the large circle = πR² = (22/7) * 14 * 14 = 616 cm²
Step 2: Find the area of one small circle:
- Radius of the small circle (r) = 3.5 cm
- Area of one small circle = πr² = (22/7) * 3.5 * 3.5 = 38.5 cm²
- Since there are two small circles, the total area of both small circles = 2 * 38.5 cm² = 77 cm²
Step 3: Find the area of the rectangle:
- Length of the rectangle = 3 cm
- Breadth of the rectangle = 1 cm
- Area of the rectangle = length * breadth = 3 cm * 1 cm = 3 cm²
Step 4: Find the area of the remaining sheet:
- Area of the remaining sheet = Area of large circle – (Area of two small circles + Area of rectangle) = 616 cm² – (77 cm² + 3 cm²) = 616 cm² – 80 cm² = 536 cm²
The area of the remaining sheet is 536 cm².
11. A circle of radius 2 cm is cut out from a square piece of an aluminium sheet of side 6 cm. What is the area of the left over aluminium sheet? (Take π = 3.14)
Ans :
1. Area of the square sheet:
- Side of the square = 6 cm
- Area of the square = side * side = 6 cm * 6 cm = 36 cm²
2. Area of the circle cut out:
- Radius of the circle = 2 cm
- Area of the circle = π * radius² = 3.14 * (2 cm)² = 12.56 cm²
3. Area of the leftover aluminium sheet:
- Area of the leftover sheet = Area of the square – Area of the circle = 36 cm² – 12.56 cm² = 23.44 cm²
Therefore, the area of the leftover aluminium sheet is 23.44 cm².
12. The circumference of a circle is 31.4 cm. Find the radius and the area of the circle. (Take π = 3.14)
Ans :
Circumference of the circle = 31.4 cm
π = 3.14
Solution:
1. Finding the radius:
- Circumference of a circle = 2πr
- 31.4 cm = 2 * 3.14 * r
- r = 31.4 cm / (2 * 3.14)
- r = 5 cm
2. Finding the area:
- Area of a circle = πr²
- Area = 3.14 * (5 cm)²
- Area = 78.5 cm²
Therefore, the radius of the circle is 5 cm and the area of the circle is 78.5 cm².
13. A circular flower bed is surrounded by a path 4 m wide. The diameter of the flower bed is 66 m. What is the area of this path? (Take π = 3.14)
Ans :
Diameter of the flower bed
= 66 m
Width of the path = 4 m
Solution:
Step 1:
Radius of the flower bed (r) = Diameter / 2 = 66 m / 2 = 33 m
Radius of the outer circle (R) = Radius of flower bed + width of the path = 33 m + 4 m = 37 m
Step 2:
Area of a circle = π * radius²
Area of the outer circle = 3.14 * (37 m)² = 4298.66 m²
Step 3:
Area of the flower bed = 3.14 * (33 m)² = 3419.46 m²
Step 4:
Area of the path = Area of the outer circle – Area of the flower bed = 4298.66 m² – 3419.46 m² = 879.2 m²
Therefore, the area of the path is 879.2 m².
14. A circular flower garden has an area of 314 m2. A sprinkler at the centre of the garden can cover an area that has a radius of 12 m. Will the sprinkler can water the entire garden?
[Take π = 3.14]
Ans :
Area of the circular flower garden
= 314 m²
Radius of the sprinkler’s coverage = 12 m
We need to find if the sprinkler can cover the entire garden.
Solution:
Step 1: Find the area covered by the sprinkler:
- Area of a circle = π * radius²
- Area covered by the sprinkler = 3.14 * (12 m)² = 3.14 * 144 m² = 452.16 m²
Step 2: Compare the areas:
Area of the garden
= 314 m²
Area covered by the sprinkler
= 452.16 m²
Since the area covered by the sprinkler is greater than the area of the garden, the sprinkler can water the entire garden.
15 . Find the circumference of the inner and the outer circles, shown in the given figure. (Take π = 3.14)
Ans :
Given:
- The radius of the outer circle is 19 meters.
- The difference between the radii of the outer and inner circles is 10 meters.
Solution:
1. Circumference of the outer circle:
Radius of the outer circle (r₁)
= 19 m
Circumference = 2 * π * r₁ = 2 * 3.14 * 19 m = 119.32 m
2. Circumference of the inner circle:
Radius of the inner circle (r₂) = 19 m – 10 m = 9 m
Circumference = 2 * π * r₂ = 2 * 3.14 * 9 m = 56.52 m
Therefore, the circumference of the outer circle is 119.32 meters, and the circumference of the inner circle is 56.52 meters.
16. How many times a wheel of radius 28 cm must rotate to go 352 m? (Take =22/7)
Ans :
Radius of the wheel = 28 cm
Total distance to cover = 352 m
Solution:
Step 1: Convert units:
- Since the radius is in centimeters and the distance is in meters, let’s convert the radius to meters for consistency.
- 1 meter = 100 centimeters
- So, radius = 28 cm / 100 = 0.28 m
Step 2: Calculate the circumference of the wheel:
- Circumference = 2 * π * radius
- Circumference = 2 * (22/7) * 0.28 m = 1.76 m
Step 3: Find the number of rotations:
- Number of rotations
= Total distance / Circumference of the wheel
- Number of rotations = 352 m / 1.76 m/rotation = 200 rotations
Therefore, the wheel must rotate 200 times to cover a distance of 352 meters.
17. The minute hand of a circular clock is 15 cm long. How far does the tip of the minute hand move in 1 hour? (Take π = 3.14)
Ans :
Length of the minute hand
= 15 cm
We need to find the distance covered by the tip of the minute hand in 1 hour.
Solution:
In 1 hour, the minute hand completes one full round of the clock.
The distance covered by the tip of the minute hand in 1 hour is equal to the circumference of the circle traced by the minute hand.
Calculating the distance:
Circumference of a circle
= 2 * π * radius
Circumference
= 2 * 3.14 * 15 cm
= 94.2 cm
