Question 1

The area of a circle is increased by 22 sq cm when its radius is increased by 1 cm. Find the original radius of the circle.

Accepted Answer

Let original radius be r. Then, according to the questions, π (r + 1)2 - πr2 = 22 ⇒ π x [(r + 1)2 - r2] = 22 ⇒ (22/7) x (r + 1 + r ) x (r + 1 - r) = 22 ⇒ 2r + 1 = 7 ⇒ 2r = 6 ∴ r = 6/2 = 3 cm

Question 2

Find the length of a rope by which a cow must be tethered in order that it may be able to graze an area of 154 sq m.

Accepted Answer

Length of to the rope = Radius of circle According to the question, πr2 = 154 ∴ r2 = 154 x (7/22) = 7 x 7 = 49 ∴ r = √49 = 7 m

Question 3

A man riding a bicycle, completes one lap of a circular field along its circumference at the speed of 14.4 km/h in 1 min 28 s. What is the area of the field?

Accepted Answer

The man takes 3600 s for 14.4 km The man will take 88 s for 14.4 x (88/3600) = 352/1000 km = 352 m Now, circumference of circular field = 352 m ⇒ 2πr = 352 m 2 x (22/7) x r = 352 ⇒ r = 56 m Therefore, area of the field = πr2 = (22/7) x 56 x 56 = 8 x 22 x 56 m2 = 9856 sq m.

Question 4

A person observed that he required 30 s time to cross a circular ground along its diameter than to cover it once along the boundary. If his speed was 30m/min, than the radius of the circular ground is (take π = 22/7)

Accepted Answer

Let the radius of circular field = r m. Speed of person in m/s = 30/60 = 1/2m/s According to the question, [(2πr) /(1/2)] - [(2r)/(1/2)] = 30 ⇒ 4πr - 4r = 30 ⇒ [4 x (22/7) - 4]r =30 ⇒ (125 - 4)r = 30 ⇒ (8.5)r = 30 ⇒ r = 30/8.5 = 3.5 m

Question 5

The area of a rectangular field is 15 time the sum of its length and breadth. If the length of that field is 40 m, what is the breadth of that field?

Accepted Answer

Length ot rectangle = 40 m Let breadth of = k Then, according of the question, (40 + k)15 = 40k ⇒ 600 + 15k = 40k ⇒ 25k = 600 ∴ k = 24 m

Question 6

The Different between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is

Accepted Answer

Let the length of rectangle = L m ∴ Breadth of rectangle = B m Using conditions from the question, L - B = 23 ....(i) 2(L + B) = 206 L + B = 103 ....(ii) On adding Eqs. (i) and (ii), we get 2L = 126 ⇒ L = 63 m ⇒ B = 103 - 63 = 40 m Then , area of rectangle = L x B = 63 x 40 = 2520 m2 .

Question 7

The area of rectangle whose length is 5 more than twice its width is 75 sq units. What is the perimeter of the rectangle?

Accepted Answer

Let the width of the rectangle = k units ∴ Length = (2k + 5) units. According to the question, Area = k(2k + 5) ⇒ 75 = 2k2 + 5k ⇒ 2k2 + 5k - 75 = 0 ⇒ 2k2 + 15k - 10k - 75 = 0 ⇒ k(2k + 15) - 5(2k + 15) = 0 ⇒ (2k + 15) (k - 5) = 0 ⇒ k = 5 and -15/2 Width cannot be negative. ∴ Width = 5 units ∴ Length = 2x + 5 = 2 x 5 + 5 = 15 unit ∴ perimeter of the rectangle = 2(15 + 5) = 40 units

Question 8

A rectangle has 30 cm as its length and 720 sq cm as its area. Its area is increased to 1 1/ 4 times its original area by increasing only its length. Its new perimeter is

Accepted Answer

Original breadth of rectangle = 720/30 = 24 cm Now , area of rectangle = (5/4) x 720 = 900 cm2 ∴ New length of rectangle = 900/24 = 37.5 cm ∴ New perimeter of rectangle = 2(l+ b) = 2(37.5 + 24 ) = 2 x 61.5 = 123 cm

Question 9

The area of a trapezium is 384 sq.cm 2. If its parallel sides are in ratio 3 : 5 and the perpendicular distance between them be 12 cm, The smaller of parallel sides is?

Accepted Answer

Let the sides of trapezium be 5k and 3k, respectively According to the question, (1/2) x [(5k + 3k) x 12] = 384 ⇒ 8k = (384 x 2)/12 = 64 ⇒ k = 64/8 = 8 cm Length of smaller of the parallel sides = 8 x 3 = 24 cm

Question 10

The side of a square is 5 cm which is 13 cm less than the diameter of a circle. What is the approximate area of the circle?

Accepted Answer

Diameter of the circle = 13 + 5 = 18 cm ∴ Radius = Diameter/2 =18/2 = 9 cm Area of the circle = πr2 = (22/7) x 92 = (22 x 81)/9 = 1782/7 = 254.57 sq cm = 255 sq cm

Question 11

If area of a square is 64 sq cm, then find the area of the circle formed by the same perimeter.

Accepted Answer

Area of square = 64 sq cm (Side)2 = 64 ∴ Side = √64 = 8 cm According to the question, ⇒ 2πr = 4 x 8 ⇒ r = (4 x 8)/2π ⇒ r = 16/π ∴ Area of the circle = π x (16/π) x (16/π) sq cm = 256/π sq cm.

Question 12

The circumference of a circle is 25 cm. Find the side of the square incribed in the circle.

Accepted Answer

Circumference of the circle = 2πr = 25 ⇒ r = 25 / (2π) According to the formula, Side of inscribed square = r√2 = 25 / (2πr x √2) = 25 / (π√2 ) cm

Question 13

The altitude of an equilateral triangle of side 2 √ 3 cm is?

Accepted Answer

∵ 1/2 x 2 √ x h = √3/4 x (2 √3)2 ∴ h = 3 cm.

Question 14

The radius of a circle is increased so that its circumference increase by 5%. The area of the circle will increase by?

Accepted Answer

Let circumference = 100 cm . Then, ∵ 2πr = 100 ⇒ r = 100/2π =50/π ⇒ New circumference = 105 cm Then, 2πR = 105 ⇒ R = 105 / (2π) &rArr Original area = [ π x (50/π) x (50/π) ] = 2500/π cm2 ⇒ New Area = [π x (105/2π) x (105/2π)] = 11025 / (4π) cm2 ⇒ Increase in area = [11025/(4π)] - 2500/π cm2 = 1025 / 4π cm2 Required increase percent [1025/(4π)] x 2500/π x 100 = 41/4% = 10.25%

Question 15

The length of minute hand of a wall clock is 7 cms. The area swept by minute hand in 30 minutes is?

Accepted Answer

Angle swept in 30 min= 180° Area swept = [(22/7) x 7 x 7] x [180°/360°] cm2 = 77 cm2

Question 16

The area of circle inscribed in an equilateral triangle is 462 cm . The perimeter of the triangle is?

Accepted Answer

∵ 22/7 x r2 = 462 ⇒ r2 = (462 x 7) /22 = 147 ⇒ r = 7√3 cm ∴ Height of the triangle = 3r = 21√3 cm Now, ∵ a2 = a2/4 + (3r)2 ⇒ 3a2/4 = (21√3)2 ⇒ a2 = (1323 x 4)/3 ⇒ a = 21x 2 = 42 cm ∴ Perimeter = 3a = 3 x 42 =126 cm

Question 17

A park is in the form of a square one of whose sides is 100 m . The area of the park excluding the circular lawn in the centre of the park is 8614 m 2. The radius of the circular lawn is?

Accepted Answer

Area of park = 100 x 100 = 10000 m2 Area of circular lawn = Area of park - area of park excluding circular lawn = 10000 - 8614 = 1386 Now again area of circular lawn = (22/7) x r2 = 1386 m2 ⇒ r2 = (1386 x 7) / 22 = 63 x 7 = 3 x 3 x 7 x 7 ∴ r = 21 m

Question 18

A room 5.44 m x 3.75 m is to be paved with square tiles. the least number of tiles required to cover the floor is?

Accepted Answer

Area of the room =(544 x 374) cm2 size of largest square tile = H.C.F. of 544 & 374 = 34 cm Area of 1 tile = (34 x 34) cm2 ∴ Least number of tiles required = (544 x 374) / (34 x 34) = 176

Question 19

The ratio of the corresponding sides of two similar triangles is 3 : 4, The ratio of their areas is?

Accepted Answer

Ratio of similar triangle = Ratio of the square of corresponding sides = (3x)2 / (4x)2 = 9x2 / 16x2 = 9/16 = 9 : 16

Question 20

A polygon has 44 diagonals the number of its sides is?

Accepted Answer

Let there be n sides of the polygon. Then it has n vertices. The total number of straight lines obtained by joining n vertices by talking 2 at a time is nC2 These nC2 lines also include n sides of polygon. Therefore, the number of diagonals formed is nC2 - n. Thus, nC2 - n = 44 ⇒ [n(n - 1)/2] - n = 44 ⇒ ( n2 - 3n) / 2 = 44 ⇒ n2 - 3n = 88 ⇒ n2 - 3n - 88 = 0 ⇒(n - 11) (n + 8) = 0 ∴ n = 11

Area Questions