**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 cm****2****, 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 m****2****. (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 m****2****. 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