# Frank Solutions for Chapter 24 Perimeter and Area Class 9 Mathematics ICSE

### Exercise 24.1

1. Find the area of a triangle whose base is 3.8 cm and height is 2.8 cm.

Given

Base of a triangle = 3.8 cm

Height of a triangle = 2.8 cm

We know that,

Area of a triangle = (1/2)× Base×Height

Substituting the values, we get,

= (1/2) × 3.8×2.8

= 5.32 cm2

Therefore, the area of a triangle is 5.32 cm2

2. Find the area of a triangle whose sides are 27 cm, 45 cm and 36 cm.

Let the three sides of a triangle be,

a = 27 cm, b = 45 cm and c = 36 cm

Semi-perimeter of a triangle = s = (a + b + c)/2

s = (27 + 45 + 36)/2

We get,

s = 54 cm

Area of a triangle = √s (s – a) (s – b) (s – c)

= √54 (54 – 27) (54 – 45) (54 – 36)

= √54×27×9 ×18

This can be written as,

= √6×9×3×9×9×6×3

= √6×6 ×3×3×9×9×9

On calculating further, we get,

= 6×3×9×3

We get,

= 486 cm2

Hence, the area of a triangle is 486 cm2

3. Find the area of an equilateral triangle of side 20 cm.

Given

Side of an equilateral triangle = 20 cm

Area of a triangle = (√3/4) ×(side)2

= (√3/4)×20×20

We get,

= 100√3 cm2

Therefore, the area of an equilateral triangle of side 20 cm is 100√3 cm2

4. Find the perimeter of an equilateral triangle whose area is 16√3 cm.

We know that,

Area of an equilateral triangle of side ‘a’ is,

A = (√3/4) a2

Given,

A = 16√3

⇒ 16√3 = (√3/4) a2

⇒ 16 = (a2/4)

⇒ a2 = 4×16

We get,

a = 2× 4

⇒ a = 8

Hence, side of an equilateral triangle is 8 cm

The perimeter of an equilateral triangle of side a = 3a

= 3×8

= 24 cm

Therefore, the perimeter of an equilateral triangle of side 8 cm is 24 cm

5. Find the area of an equilateral triangle having perimeter of 18 cm.

We know that,

Perimeter of an equilateral triangle (P) of side a = 3a

Here,

P = 18 cm

Side of the equilateral triangle = 6 cm

Area of an equilateral triangle (A) of side ‘a’ is A = (√3/4) a2

A = (√3/4)× 62

⇒ A = (√3/4)×36

We get,

A = 9√3

Hence, area of an equilateral triangle (A) of side 6 cm is 9√3 cm2

6. The side of a square is of length 20 mm. Find its perimeter in cm.

Given

Side of a square = 20 mm

We know,

Perimeter of square = 4× side

Perimeter of square = 4× 20 mm

We get,

Perimeter of square = 80 mm

Perimeter of square = (80/10) cm

Perimeter of square = 8 cm

Therefore, the perimeter is 8 cm

7. The area of a square is 36 cm2. How long are its sides?

Area of a square = 36 cm2

(Side)2 = 36 cm2

Side = √36 cm

We get,

Side = 6 cm

Therefore, the length of each side is 6 cm

8. The sides of a rectangle are 5 cm and 3 cm respectively. Find its area in mm2

Given

Length of a rectangle = 5 cm

Breadth of a rectangle = 3 cm

We know that,

Area of a rectangle = Length×Breadth

Area of a rectangle = 5 cm×3 cm

Area of a rectangle = 15 cm2

Area of a rectangle = 15× 100 mm2

Area of a rectangle = 1500 mm2

9. Find the area and perimeter of the given figure.

= AB x BC + (DE)2

On substituting, we get,

= 8 cm×3 cm + (3 cm)2

= 24 cm2 + 9 cm2

= 33 cm2

Perimeter of the given figure = AB + BC + CD + DE + EF + FG + GH + HA

Perimeter of the given figure = AB + BC + (CD + EF + GH) + DE + FG + HA

Perimeter of the given figure = 8 + 3 + 8 + 3 + 3 + 3

We get,

Perimeter of the given figure = 28 cm

Therefore, the area and perimeter of the given figure is 33 cm2 and 28 cm

10. Find the shaded area in the given figure.

Area of shaded region = Area of rectangle PQRS – (Area of rectangle ABFG + Area of square CDEF)

= PQ x QR – [(AB x AG) + (CD)2]

= (8 x 9) cm2 – [(2×4) cm2 + (2)2 cm2]

= 72 cm2 – [8 cm2 + 4 cm2]

= 72 cm2 – 12 cm2

We get,

= 60 cm2

Therefore, the area of shaded region in the given figure is 60 cm

11. Find the area and perimeter of the circles with the following:

(i) Radius = 2.8 cm

(ii) Radius = 10.5 cm

(iii) Diameter = 77 cm

(iv) Diameter = 35 cm

(i) We know,

The area of a circle with radius r = 𝝅r2

Hence,

The area of a circle with radius 2.8 cm = 𝝅 (2.8)2

= (22/7) (2.8)2

We get,

= 24.64 cm2

The circumference of a circle with radius r = 2 𝝅r

The circumference of a circle with radius 2.8 cm = 2×𝝅×2.8

= 2×(22/7)×2.8

We get,

= 17.6 cm

(ii) The area of a circle with radius r = 𝝅r2

Therefore,

The area of a circle with radius 10.5 cm = 𝝅(10.5)2

= (22/7) (10.5)2

We get,

= 346.5 cm2

The circumference of a circle with radius r = 2𝝅r

The circumference of a circle with radius 10.5 cm = 2×𝝅 ×10.5

= 2× (22/7)×10.5

We get,

= 66 cm

(iii) The radius of a circle with diameter d is r = (d/2)

The area of a circle with radius r = 𝝅r2

The radius of a circle with diameter 77 is r = (77/2)

r = 38.5 cm

The area of a circle with radius r = 𝝅 (38.5)2

r = (22/7) ×(38.5)2

r = 4658.5 cm2

The circumference of a circle with diameter d is 𝝅d

The circumference of a circle with diameter 77 is 𝝅 × 77

= (22/7)×77

We get,

= 242 cm

(iv) The radius of a circle with diameter d is r = (d/2)

The area of a circle with radius r = 𝝅r2

The radius of a circle with diameter 35 is r = (35/2) = 17.5 cm

The area of a circle with radius r = 𝝅 (17.5)2

= (22/7)×(17.5)2

We get,

= 962.5 cm2

The circumference of a circle with diameter d is 𝝅d

The circumference of a circle with diameter 35 is 𝝅×35 = (22/7)×35

= 110 cm

12. Find the area and perimeter of the following semicircles:

(i) Radius = 1.4 cm

(ii) Diameter = 7 cm

(iii) Diameter = 5.6 cm

(i) The area of a semi-circle with radius r = (𝝅r2)/2

The perimeter of a semi-circle with radius r = 𝝅r + 2r

= r (𝝅 + 2) {By taking ‘r’ as common}

= r {(22/7) + 2}

We get,

= (36/7)×r

Given radius = 1.4 cm

The area of a semi-circle with radius 1.4 cm = {𝝅×(1.4)2/2}

= {(22/7)×(1.4)2/2}

= 3.08 cm2

The perimeter of a semi-circle with radius r = 𝝅r + 2r

= 1.4 (𝝅 + 2)

= 1.4 {(22/7) + 2}

= (36/7)×1.4

We get,

= 7.2 cm

(ii) The radius of a circle with diameter d is r = (d/2)

The area of a semi-circle with radius r = 𝝅r2/2

The perimeter of a semi-circle with radius r = 𝝅r + 2r

= r (𝝅 + 2)

= r {(22/7) + 2}

= (36/7)× r

The radius of a circle with diameter 7 is r = (7/2)

= 3.5 cm

The area of a semi-circle with radius 3.5 = (3.5)2/2

= (22/7)×(3.5)2/2

We get,

= 19.25 cm2

The perimeter of a semi-circle with radius r = 𝝅×3.5 + 2×3.5

= 3.5 (𝝅 + 2)

= 3.5 {(22/7) + 2}

= (36/7)×3.5

We get,

= 18 cm

(iii) The radius of a circle with diameter d is r = (d/2)

The area of a semi-circle with radius r = (𝝅r2/2)

The perimeter of a semi-circle with radius r = 𝝅r + 2r

= r (𝝅 + 2)

= r {(22/7) + 2}

= (36/7)× r

The radius of a circle with diameter 5.6 is r = (5.6)/2

= 2.8 cm

The area of a semi-circle with radius 2.8 = {(2.8)2}/2

We get,

= 12.32 cm2

The perimeter of a semi-circle with radius r = 𝝅×2.8 + 2×2.8

= 2.8 (𝝅 + 2)

= 2.8 {(22/7) + 2}

= (36/7)×2.8

We get,

= 14.4 cm

13. Find the area of a circular field that has a circumference of 396 m.

The circumference of a circle with radius r = 2 𝝅r

Here,

Given circumference of a circle = 396 m

2 𝝅r = 396

⇒ r = 396/2 𝝅

⇒ r = (396×7)/(2×22)

We get,

r = 2772/44

⇒ r = 63 m

We know,

The area of a circle with radius r = 𝝅r2

Hence,

The area of a circle with radius 63 m = (63)2 = (22/7)×(63)2

= (22/7)×3969

We get,

= 12, 474 m2

14. Find the circumference of a circle whose area is 81 𝝅 cm2

The area of a circle with radius r = 𝝅r2

Here,

Given area of a circle = 81 𝝅 cm2

81 𝝅 = 𝝅r2

⇒ r2 = 81

We get,

r = 9 cm

The circumference of a circle with radius r = 2 𝝅r

The circumference of a circle with radius 9 = 2 𝝅×9

We get,

= 18 𝝅 cm

15. The circumference of a circle exceeds its diameter by 450 cm. Find the area of the circle.

Let the radius of a circle = r cm

Circumference of a circle = 2 𝝅r cm

Diameter of a circle = 2r cm

Given,

Circumference of a circle – Diameter of a circle = 450 cm

2 𝝅r – 2r = 450

Taking 2r common we get,

2r (𝝅 – 1) = 450

⇒ 2r {(22/7) – 1} = 450

⇒ 2r (15/7) = 450

⇒ r = (450×7)/(2×15)

⇒ r = 3150/30

We get,

r = 105 cm

Hence,

Area of a circle = 𝝅r2 = (22/7)×105×105

On calculating, we get,

= 34650 cm2

16. (a) Find the area of each of the following figure:

(b) Find the area of each of the following figure:

(c) Find the area of each of the following figure:

(d) Find the area of each of the following figure:

(a)

(b)

(c)

(d)

17. Find the area of a parallelogram whose base is 12 cm and the height is 5 cm.

Area of a parallelogram with base b and height h is A = b × h

∴ Area of a parallelogram with base 12 cm and height 5 cm is A = 12 × 5 = 60 cm2

18. Find the height of a parallelogram whose area is 144 cm2 and the base is 18 cm.

19. Find the area of quadrilateral, whose diagonals of lengths 18 cm and 13 cm intersect each other at right angle.

When two diagonals of a quadrilateral intersect each other at right angles.

Area of quadrilateral = 1/2 × Product of the diagonals

∴ Area of required quadrilateral = 1/2 × 18 × 13

= 117 cm2

20. In a rectangle ABCD, AB = 7 cm and AD = 25 cm. Find the height of a triangle whose base is AB and whose area is two times the area of the rectangle ABCD.

21. Two adjacent sides of a parallelogram are 34 cm and 20 cm. If one of its diagonal is 42 cm. Find:

(a) area of the parallelogram

(b) distance between its shorter sides

(a)

(b)

22. One side of a parallelogram is 12 cm and the altitude corresponding to it is 8 cm. If the length of the altitude corresponding to its adjacent side is 16 cm, find the length of the adjacent side.

23. Two adjacent sides of a parallelogram are 20 cm and 18 cm. If the distance between the larger sides is 9 cm, find the area of the parallelogram. Also, find the distance between the shorter sides.

24. Find the perimeter and area of a rectangle whose length and breadth are 12 cm and 9 cm respectively.

25. The area of a floor of a rectangular room is 360 m2. If its length is 24 cm, find its perimeter.

26. The length of a rectangular field is thrice of its width. If the perimeter of this field is 1.6 km, find its area in sq. m.

27. A rectangular field 240 m long has an area 36000 m2. Find the cost of fencing the filed at Rs 2.50 per m.

28. In a trapezium the parallel sides are 12 cm and 8 cm. If the distance between them is 6 cm. find the area of the trapezium.

29. A rectangular floor 45 in long and 12 m broad is to be paved exactly with square tiles, of side 60 cm. Find the total number of tiles required to pave it. If a carpet is laid on the floor such as a space of 50 cm exists between its edges and the edges of the floor, find what fraction of the floor is uncovered.

30. A rectangular hall of 40 m by 24 m is covered with carpets of size 6 m × 4m. Find the number of carpets required to cover the hall.

31. The area of a square garden is equal to the area of a rectangular plot of length 160 m and width 40m. Calculate the cost of fencing the square garden at Rs 12 per m.

32. Find the area of a square whose diagonals is 12√2 cm.

33. Find the perimeter and area of a square whose diagonals is 5√2 cm. Give your answer correct to two decimal places if √2 = 1.414.

34. Find the perimeter of a rhombus whose diagonals are 24 cm and 10 cm.

35. The perimeter of a square plot of land is 64 m. The area of a nearby rectangular plot is 24m2 more than the area of the given square. If the length of the rectangular is 14m, find its breadth.

36. The perimeter of a square is 128 cm and that of another is 96 cm, Find the perimeter and the diagonals of a square whose area is equal to the sum of the areas of these two squares.

37. The area of a square plot of side 80 m is equal to the area of a rectangular plot of length 160m. Calculate the width of the rectangular plot and the cost of fencing it Rs 7.50 per m.

38. The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 cm2. Find the dimensions of the squares.

39. Find the area of a rhombus, whose one side and one diagonal measure 20 cm and 24 cm respectively.

40. The area of a rhombus is 234 cm2. If its one diagonal is 18 cm, find the lengths of its side and the other diagonal. Also, find perimeter of the rhombus.

41. Find the area of a rhombus whose perimeter is 260 cm and the length of one of its diagonal is 66 cm.

42. A rectangular field is 80 m long and 50 m wide. A 4 m wide road runs through the centre of the field parallel to the length and breadth of the field. Find the total area of the roads.

43. A quadrilateral filed of unequal sides has a longer diagonal with 140 m. the perpendiculars from opposite vertices upon this diagonals are 20 m and 14 m. find the area of the field.

44. A lawn in the shape of a rectangle is to be developed in front of a Marriage Hall. The length and breadth of the lawn are 44 m and 32 m. A space of 2 m is left on the two shorter sides and one longer side for flowers and in the remaining area grass is laid. Calculate the area of the flower space and the area on which grass is laid.

45. The floor of a room is size 6 m × 5 m. Find the cost of covering the floor of the room with 50 cm wide carpet at the rate of Rs 24.50 per metre. Also, find the cost of carpeting the same hall if the carpet, 60 cm, wide, is at the rate of Rs 26 per metre.

46. A rectangular field is 240 m long and 180 m broad. In one corner a farm house is built on a square plot of side 40 m. Find the area of the remaining portion and the cost of fencing the open sides at Rs 25 per m.

47. Inside a square filed of side 44m, a square flower bed is prepared leaving a gravel path all round the flower bed. The total cost of laying the flower bed at Rs 25 per sq m and gravelling the path at Rs 120 per sq m is Rs 80320. Find the width of the gravel path.

48. The length and breadth of a rectangular filed are in the ratio 8 : 5. A 2 m wide path runs all around outside the filed. The area of the path is 848 m2. Find the length and breadth of the filed.

49. A footpath of uniform width runs all around the inside of a rectangular garden of 40 m × 30 m. If the path occupies 136 m2. Find the width of the path.

50. How many tiles, each of area 625 cm2, will be needed to pave a footpath to pave a footpath which is 1 m wide and surrounds a grass plot of size 38 m × 14 m?

51. PQRS is a square with each side 6 cm. T is a point on QR such that the (area of PQT)/(area of trapezium PTRS) = 1/3. Find the length of TR.

52. The perimeter of a rectangular plot is 300 m. It has an area of 5600 m2. Taking the length of the plot as x m, calculate the breadth of the plot in terms of x, form an equation and solve it to find the dimensions of the plot.

53. Find the area of a quadrilateral filed whose sides are 12m, 9 m, 18 m and 21 m respectively and the angle between the first two sides is a right angle. Take the value of √6 as 2.5.

54. Find the diagonal of a quadrilateral whose area is 756 cm2 and the perpendiculars from opposite vertices are 17 cm and 19 cm.

55. A wire when bent in the form of a square encloses an area of 16 cm2. Find the area enclosed by it when the same wire is bent in the form of

(a) a rectangular whose sides are in the ratio of 1 : 3.

(b) an equilateral triangle

(a)

(b)

56. A chessboard contains 64 equal squares and the area of each square is 6.25 cm2. A 2 cm wide border is left inside of the board . Find the length of the side of the chessboard.

57. The cross-section of a canal is a trapezium in shape. If the canal is 10m wide at the top, 6 m wide at bottom and the area of cross-section is 72 sq. m, determine its depth.

### Exercise 24.2

1. Find the area and perimeter of the circles with the following:

(i) Radius = 2.8 cm

(ii) Radius = 10.5 cm

(iii) Diameter = 77 cm

(iv) Diameter = 35 cm

(i)

(ii)
(iii)
(iv)

2. Find the area and perimeter of the following semicircles:

(i) Radius = 1.4 cm

(ii) Diameter = 7 cm

(iii) Diameter = 5.6 cm

(i)

(ii)
(iii)

3. Find the area of a circular field that has a circumference of 396 m.

4. Find the circumference of a circle whose area is 81
π cm2.

5. The diameter of a wheel is 1.4 m. How many revolutions does it make in moving a distance of 2.2 km?

6. The circumference of a circle exceeds its diameter by 450 cm. Find the area of the circle.

7. The circumference of a circle is numerically equal to its area. Find the area and circumcircle of the circle.

8. The sum of circumference and diameter of a circle is 176 cm. Find the area of the circle.

9. Find the radius and area of the circle which has circumference equal to the sum of circumferences of the two circles of radii 3 cm and 4 cm respectively.

10. The diameter of two circles are 28 cm and 24 cm. Find the circumference of the circle having its area equal to sum of the areas of the two circles.

11. The radii of two circles are in the ratio 5 : 8. If the difference between their areas is 156p cm2, find area of the bigger circle.

12. The diameter of three wheels are in the ratio 2 : 4 : 8. If the sum of the circumferences of these circles be 132 cm, find the difference between the areas of the largest and the smallest of these wheels.

13. The wheel of a car makes 10 revolutions per second. If its diameter is 70 cm, find the speed of the car in km per hr.

14. The speed of a car is 66 km per hour. If each wheel of the car is 140 cm in diameter, find the number of revolutions made by each wheel per minute.

15. A cart wheel makes 9 revolutions per second. If the diameter of the wheel is 42 cm, find its speed in km per hour. (Answer correct to the nearest km).

16. A bucket is raised from a well by means of a rope wound round a wheel of diameter 35 cm. If the bucket ascends in 2 minutes with a uniform speed of 1.1 m per sec, calculate the number of complete revolutions the wheel makes n raising the bucket.

17. The circumference of a garden roller is 280 cm. How many revolutions does it make in moving 490 m ?

The circumference of a circle = 280 cm = 2.8 m

Number of revolutions = (Total distance moved)/(Circumference of circle) = 490/2.8 = 175

18. The diameter of a cycle wheel is 4.5\11 cm. How many revolutions will it make in moving 6.3 km ?

19. The area of the circular ring enclosed between two concentric circles is 88 cm2. Find the radii of the two circles. If their differences is 1 cm.

20. Find the area enclosed between two concentric circles, if their radii are 6 cm and 13 cm respectively.

21. The area between the circumferences of two concentric circles is 2464 cm2. If the inner circle has a circumference of 132 cm, calculate the radius of the outer circle.

22. A wire when bent in the form of a square encloses an area of 484 cm2. If the same wire is bent into the form of a circle, find the area of the circle.

23. A wire bent in the form of an equilateral triangle has an area of 121√3 cm2. If the same wire is bent into the form of a circle, find the area enclosed by the wire.

24. Find the circumference of the circle whose area is 25 times the area of the circle with radius 7 cm.

25. The circumference of a circle is equal to the perimeter of a square. The area of the squares is 484 sq. m. Find the area of the circle.

26. A wire is in the form of a circle of radius 42 cm. It is bent into a square. Determine the side of the square and compare the area of the regions enclosed in the two cases.

27. A 7 m wide road surrounds a circular garden whose area is 5544 m2. Find the area of the road and the cost of tarring it at the rate of Rs 150 per m2.

28. A 4.2 m wide road surrounds a circular plot whose circumference is 176 m. Find the cost of paving the road at Rs 75 per m2.

29. Two circles touch each other externally. The sum of their areas is 58 π cm2 and the distance between their centres us 10 cm. Find the radii of the two circles.

30. The sum of diameters of two circles is 112 cm and the sum of their areas is 5236 cm2. Find the radii of the two circles.

31. The sum of the radii of two circles is 10.5 cm and the difference of their circumferences is 13.2 cm Find the radii of the two circles.

32. A lawn is in the shape of a semicircle of diameter 42m. The lawn is surrounded by a flower bed of width 7 m all around. Find the area of the flower bed in m2.

33. A square is inscribed in a circle of radius 6 cm. Find the area of the square. Give your answer correct to two decimal if √2 = 1.414.

34. The cost of fencing a circular field at the rate of Rs 250 per metre is Rs 55000. The field is to be ploughing at the rate of Rs 15 per m2. Find the cost of ploughing the field. 