Mensuration - Exercise 23(A)

Find the perimeters of:

(i) A triangle ABC having sides of lengths, AB = 7 cm, BC = 5 cm, AC = 8.5 cm.

(ii) A rectangle of length 12 cm and breadth 7.5 cm.

(iii) A square having each side of length 4.5 cm.

(iv) A rhombus having each side of length 6.3 cm.

Answer

(i) A triangle ABC having sides of lengths, AB = 7 cm, BC = 5 cm, AC = 8.5 cm.

Given:

Sides: AB = 7 cm, BC = 5 cm, AC = 8.5 cm

Perimeter of triangle = Sum of all sides

Perimeter of triangle = AB + BC + AC

= 7 cm + 5 cm + 8.5 cm

= 20.5 cm

Perimeter of triangle = 20.5 cm.

(ii) A rectangle of length 12 cm and breadth 7.5 cm.

Given:

Length = 12 cm, breadth = 7.5 cm

Perimeter of rectangle = 2(length + breadth)

= 2(12 cm + 7.5 cm)

= 2(19.5 cm)

= 39 cm

Perimeter of rectangle = 39 cm.

(iii) A square having each side of length 4.5 cm.

Given:

Side = 4.5 cm

Perimeter of square = 4 x side

= 4 x 4.5 cm $\quad$[Substituting the value]

= 18 cm

Perimeter of square = 18 cm.

(iv) A rhombus having each side of length 6.3 cm.

Given:

Side = 6.3 cm

Perimeter of rhombus = 4 x side

= 4 x 6.3 cm $\quad$[Substituting the value]

= 25.2 cm

Perimeter of rhombus = 25.2 cm.

A rectangular field has perimeter 186 m and breadth 35 m. Find its length.

Answer

Given:

Perimeter of rectangle = 186 m

Breadth = 35 m

Perimeter of rectangle = 2(length + breadth)

186 m = 2(length + 35 m)

186 m = (2 x length + 2 x 35 m)

186 m = 2 x length + 70 m

186 m - 70 m = 2 x length

116 m = 2 x length

⇒ length = $\dfrac{116}{2}$ m

⇒ length = 58 m

Length of the rectangular field = 58 m.

Find the length of each side of a square plot having perimeter 172 m.

Answer

Given:

Perimeter = 172 m

Perimeter of square = 4 x side

172 m = 4 x side

⇒ side = $\dfrac{172}{4}$ m

⇒ side = 43 m

Length of each side of the square plot = 43 m.

A rectangular plot is 125 m long and 72 m broad. Find the cost of fencing it at ₹ 27 per metre.

Answer

Given:

Length = 125 m

Breadth = 72 m

Rate = ₹ 27 per metre

Fencing is done along the boundary, so we first find the perimeter.

Perimeter of rectangle = 2(length + breadth)

= 2(125 m + 72 m)

= 2(197 m)

= 394 m

Total cost = Perimeter x Rate

= 394 m x ₹ 27

= ₹ 10638

Total cost of fencing the rectangular plot = ₹ 10638.

Find the circumference of a circle of radius 17.5 cm. $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Radius (r) = 17.5 cm

$\pi = \dfrac{22}{7}$

Circumference of a circle = 2$\pi$r

= 2 × $\dfrac{22}{7}$ × 17.5 cm

= 2 × 22 × 2.5 cm

= 110 cm

Circumference of the circle = 110 cm.

The circumference of a circular garden is 66 m. Find its diameter. $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Circumference of the circular garden = 66 m

$\pi = \dfrac{22}{7}$

Circumference of a circle = $\pi d$ $\quad$[d is diameter]

66 m = $\dfrac{22}{7}$ × d

⇒ d = $\dfrac{66 \times 7}{22}$ m

⇒ d = $\dfrac{462}{22}$ m

⇒ d = 21 m

Diameter of the circular garden = 21 m.

The diameter of a wheel of a car is 1.12 m. Find the distance covered by the car in making 500 revolutions by its wheels. $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Diameter = 1.12 m

Number of revolutions = 500

$\pi = \dfrac{22}{7}$

Distance covered in 1 revolution = Circumference of the wheel

Circumference of a circle = $\pi d$ $\quad$[d is diameter]

= $\dfrac{22}{7}$ x 1.12 m

= $\dfrac{24.64}{7}$ m

= 3.52 m

Total distance covered = Circumference x Number of Revolutions

= 3.52 m x 500

= 1760 m

Converting it into km:

1 km = 1000 m

∴ 1760 m = 1.76 km

The car covers a distance of 1.76 km.

The radius of a wheel of a cycle is 70 cm and it takes 5 minutes to make 300 rotations. Find the speed of the cycle in km per hour. $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Radius (r) = 70 cm

Time = 5 minutes

Number of rotations = 300

$\pi = \dfrac{22}{7}$

Distance covered in 1 rotation = Circumference of the wheel

Circumference of a circle = 2πr

= 2 × $\dfrac{22}{7}$ × 70 cm

= 2 × 22 × 10 cm

= 440 cm

Total distance = Circumference × number of rotations

= 440 cm × 300

= 132000 cm

Converting distance into km:

1 km = 100000 cm

∴ 132000 cm = $\dfrac{132000}{100000}$ km = 1.32 km

Converting time into hours:

1 hour = 60 minutes

∴ 5 minutes = $\dfrac{5}{60}$ hour = $\dfrac{1}{12}$ hour

Speed = $\dfrac{\text{Distance}}{\text{Time}}$

= $\dfrac{1.32}{\dfrac{1}{12}}$ km/hr

= 1.32 × 12 km/hr

= 15.84 km/hr

Speed of the cycle = 15.84 km/hr.

How many revolutions would a cycle wheel of diameter 1.6 m make to cover a distance of 352 metres? $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Diameter = 1.6 m

Total distance = 352 m

$\pi = \dfrac{22}{7}$

Distance covered in 1 revolution = Circumference of the wheel

Circumference of a circle = $\pi d \quad$ [d is diameter]

= $\dfrac{22}{7}$ × 1.6 m

= $\dfrac{35.2}{7}$ m

= 5.02 m

Number of revolutions = $\dfrac{\text{Total distance}}{\text{Circumference}}$

= $\dfrac{352}{\dfrac{35.2}{7}}$

= 352 × $\dfrac{7}{35.2}$

= 10 × 7

= 70

Number of revolutions made by the cycle wheel = 70.

A wire is bent in the form of a square of side 16.5 cm. It is straightened and then bent into a circle. What is the radius of the circle so formed? $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Side of the square = 16.5 cm

$\pi = \dfrac{22}{7}$

When a wire is reshaped, its total length remains the same.

Perimeter of square = 4 × side

= 4 × 16.5 cm $\quad$[Substituting the value]

= 66 cm

So, circumference of the circle = 66 cm

Circumference of a circle = 2πr

66 cm = 2 × $\dfrac{22}{7}$ × r

66 cm = $\dfrac{44}{7}$ × r

⇒ r = $\dfrac{66}{\dfrac{44}{7}}$ cm

⇒ r = 66 × $\dfrac{7}{44}$ cm

⇒ r = $\dfrac{21}{2}$ cm

⇒ r = 10.5 cm

Radius of the circle = 10.5 cm.

The wheel of a car rotated 1000 times in travelling a distance of 1.76 km. Find the radius of the wheel. $\Big(\text{Take } \pi = \dfrac{22}{7}\Big)$

Answer

Given:

Total distance = 1.76 km

Total rotations = 1000

Converting distance into cm:

1 km = 100000 cm

∴ 1.76 km = 1.76 × 100000 cm = 176000 cm

Distance covered in 1 rotation = Circumference of the wheel

= $\dfrac{\text{Total distance}}{\text{Total rotations}}$

= $\dfrac{176000}{1000}$ cm

= 176 cm

Circumference of a circle = 2πr

176 cm = 2 × $\dfrac{22}{7}$ × r

176 cm = $\dfrac{44}{7}$ × r

⇒ r = $\dfrac{176}{\dfrac{44}{7}}$ cm

⇒ r = 176 × $\dfrac{7}{44}$ cm

⇒ r = 28 cm

Radius of the wheel = 28 cm.