Students and Teachers Forum

1. Find the measure of base side of given square based pyramid if its area of triangular faces is 260 cm2.

Solution: Here,        Slant height(h) =13cm and length of square base = a cm Now,         The area of triangular surfaces of pyramid= 260 cm2    Or, 2×ah = 260   Or, 2×a×13 .....

2. Find the total surface area of the pyramid.

We have given,         OB = 25cm = e         OP = 24cm = l         AB = a First of all,       PB2 = OB2 – OP2 Or, PB = √[252 - 242]     .....

3. Find the radius and volume of the sphere if its surface area is 154 cm2.

Given,        Surface area of the sphere (TSA) = 154 cm2 We know,        Surface area of the sphere (TSA) = 4πr2 Or,  154 cm2 = 4⨉  3.14 ⨉ r2 Or,  154 cm2 = 12.56 ⨉ r2 Or,  .....

4. If the volume of the sphere is 4π/3 cm3, find its radius and the surface area.

Given, Volume of the sphere (V) = 43 πcm3 Or,  43r3 = 43 π cm3 Or,   r3 = 1 cm3 Or,   r = ∛(1cm3) Or,   r = 1 cm So, the radius of the sphere is 1 cm. Now, Surface area of the sphere (TSA) = .....

5. Find the volume of a sphere having radius 21 cm.

Given,         Radius of sphere (r) = 21cm.         Volume =? We have,        Volume of sphere = 43πr3                       .....

6. A solid metallic sphere of 3 cm melted to form a right circular cylinder having the same radius. Find the height of the cylinder.

Given, Radius of the metallic sphere (r) = 3 cm Then, Volume of the metallic sphere (V) = 43πr3                                           .....

7. The ratio of surface area and volume of a sphere are in the ratio 3:7 cm. What is radius of sphere?

Let r be the radius of sphere. Given,       surface area : volume=3:7 Using formula, we get, or, (4πr2):(43 πr3 ) = 3:7 or, 3r = 37 or, r = 7cm ∴   r  = 7cm. Therefore, radius of sphere is .....

8. The curved area of the sphere is 400 cm2. Find the total surface area of hemisphere formed from same sphere.

Here,         Surface area of the sphere (A) = 400cm2         Or, 4πr2 = 400 cm2.          Or, πr2  =100 cm2. Now, TSA of hemisphere = 3πr2      .....

9. The area of the great circle is 49 π cm2. Find the volume of the sphere.

Given,         Area of the great circle of the sphere (A) = 49π cm2 Or,   πr2 = 49π cm2 Or,   r2 = 49 cm2  Or,  r2 = 49 cm2 Or,  r = √(49 cm2 ) Or,  r = 7 cm So, the radius .....

10. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article.

Given,          Height of the cylinder (h) = 10 cm          Radius of the base (r) = 3.5 cm. Now,         Total surface area of the article          .....

11. Find the radius of the base and the height of the cylinder given that its volume is 770 cm and the area of its curved surface is 440 cm2.

Given,        Volume of the cylinder (V) = 770 cm3       Area of curved surface (CSA) = 440 cm2 We know,       Volume of cylinder (V) = πr2h Or,  770 cm3 = πr2h Or,  770 cm3 = πrh .....

12. 35 identical circular plates, each of thickness 0.4 cm and diameter 14 cm are placed one above another to form a right circular cylinder. Find the curved surface area of the cylinder so formed.

Here,             Total number of circular plates (N) = 35             Diameter of each plate (d) = 14 cm             Radius of each plate (r) = 7 cm   .....

13. The curved surface area of a cylinder is 462 cm2. If the height of the cylinder is 14 cm, find the area of the base of the cylinder.

Given,         Curved surface area of the cylinder (CSA) = 462 cm2         Height of the cylinder (h) = 14 cm. We know,         Curved surface area of the cylinder (CSA) = .....

14. A juice seller was serving his customers using glasses. The inner diameter of the cylinder glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm, find the apparent capacity of the glass and its actual capacity.

Given,        Diameter of glass (d) = 5 cm        Radius of glass (r)  = 2.5 cm       Height of the glass (h) = 10 cm. Now,   Apparent capacity of glass  = Volume of cylindrical .....

15. A cylinder of volume 7,70,000 cm3 is cut vertically along its diameter measuring 14 cm. Find the total surface area of each half so formed.

Given,         Volume of the object (V) = 7,70,000 cm3         Diameter of the object (d) = 14 cm         Radius of the object (r) =7 cm. We know,       Volume of the .....

16. A solid object is formed by cutting a cylinder vertically along the diameter measuring 24 cm. If the height of the object so formed is 12 cm, find the volume of the object.

Given,        Diameter of the object (d) = 24 cm        Radius of the object (r) =  12 cm        Height of the object (h) = 12 cm. We know,        Volume of the .....

17. The volume of a solid object formed by cutting a cylinder vertically along the diameter is 1,234 cm3. If the radius of the solid object is 14 cm, find the height of the object.

Given, Volume of the solid object (V) = 1234 cm3 Radius of the solid object (r) = 14 cm We know, Volume of the half of cylinder (V) = 12πr2h Or, 1234 cm3 = 2214(14cm)2 ⨉ h     Or, 1234 cm3 = 308 cm2 ⨉ h  Or, h = 4 cm. So, .....

18. A solid cylinder has a length of 7 cm. It is cut into two halves by cutting vertically along the diameter. If the total surface area of each half is 406 cm2, find the radius of the cylinder.                                               

Given,       Length of the cylinder (h) = 7 cm       Total surface area of each half (TSA) = 406 cm2       Radius of the cylinder (r) = ? We know,        Total surface area of half of .....

19. A solid object is formed by cutting a cylinder vertically along the diameter measuring 49 cm. If the volume of the object so formed is 3,773 cm3, find the height of the object.

Given,          Diameter of the object (d) = 49 cm          Radius of the object (r) = d2 =  24.5 cm          Volume of the object (V) = 3,773 cm3. We know,   .....

20. A solid cylinder is cut into two halves cutting vertically along the diameter measuring 28 cm. If the total surface area of each half is 1,624 cm2, find the length of the cylinder.

Given,          Diameter of the cylinder (d) = 28 cm          Radius of the cylinder (r)  = 14 cm          Total surface area of each half (TSA) = 1624 cm2     .....

21. The radius of the base of the cylinder is 35 cm and height is 60 cm, find the
(a) the area of the curved surface
(b) the total surface area
(c) volume of the cylinder.

Given,       Radius of the base of the cylinder (r) = 35 cm       Height of the cylinder (h) = 60 cm. We know, (i) Area of the curved surface of cylinder (CSA) = 2πrh               .....

22. Find the curved surface area, total surface area and volume of the given cylinder. (Fig. cylinder with area of base = 38.5 cm2 and height 8 cm)

Given,      Area of the base of cylinder (A)     = 38.5 cm2 Or, πr2           = 38.5 cm2 Or, 3.14 ⨉ r2 = 38.5 cm2 Or, r2             = 12.25 .....

23. The total cost of a pipe 14 cm long is Rs. 154 at the rate of Rs. 7 per cm. If the internal radius of the pipe is 5 cm, find the thickness of the pipe.

Given,      Length of the pipe (h) = 14 cm      Rate of pipe (C) = Rs.7 per cm3      Total cost of pipe (T) = Rs. 154. We know,       Volume of pipe (V)  = TC       .....

24. The volume of the cylinder is 7700 cm3, whose area of the base is 3850 cm2. Find the total surface area of the cylinder.

Given,         Volume of the cylinder (V)     = 7700 cm3         Area of the base (A)     = 3850 cm2. We know,         Volume of the cylinder (V)   .....

25. A cylindrical water pipe is 126 cm long. Its outer and inner diameters are 8 cm and 5 cm respectively. Find the volume of the pipe.

Given,       Length of the water pipe (h) = 126 m       Outer diameter of the pipe (d1) = 8 cm       Outer radius of the pipe (r1)  = 4 cm       Inner diameter of the pipe (d2) = 5 .....

26. A cylinder of volume 4,400 cm3 is cut vertically along its diameter. If the cylinder measures 14 cm in length, find the total surface area of each half so formed.

Given,         Volume of the object (V) = 4,400 cm3         Length of the object (h) = 14 cm. We know,         Volume of the cylindrical half (V) = ½ ⨉ πr2h   Or, 4400 cm3 .....

27. Find the curved surface area and volume of the cylinder whose diameter is 14 cm and height is 30 cm.

Given,        Diameter of the cylinder (d) = 14 cm        Radius of the cylinder (r) = 7 cm        Height of the cylinder (h) = 30 cm. We know, Curved surface area of cylinder (CSA) = .....

28. How many cubic meters of earth must be dug out to sink a cylindrical well 20 m deep and 3.5 m in diameter. What volume if water will it contain? (1 cu. m = 1 kl)

Given,         Depth of the cylindrical well (h) = 20 m         Diameter of the cylindrical well (d) = 3.5 m         Radius of the cylindrical well (r) = d2         .....

29. The curved surface area of the cylinder whose radius and height are equal is 308 cm2. Find the total surface area of the cylinder.

Given,         Curved surface area of the cylinder (CSA) = 308 cm         Radius of the cylinder (r) = Height of the cylinder (h). We know,          Curved surface area of cylinder .....

30. The volume of a cylindrical can is 1.54 liters. If the area of its base is 77 cm2, find its height.

Given,     Volume of cylindrical can (V) = 1.54 l                                                      = .....

31. An iron pipe is 140 cm long. If its internal diameter is 5 cm and the thickness of the iron is 1 cm, find the volume of iron in the pipe.

Given,         Length of the iron pipe (h) = 140 cm         Internal diameter of the iron pipe (d2) = 5 cm        Or, internal radius of the iron pipe (r2) = d22 = 2.5 cm     .....

32. 35 identical cylindrical plates each of thickness 0.4 cm and diameter 14 cm are placed one above another to form a right circular cylinder. Find the total surface area and the volume of the cylinder so formed.

Given,        Thickness of the cylindrical plates (t) = 0.4 cm        Total number of cylindrical plates (N) = 35        So, total height of the cylinder so formed (h) = N ⨉ t   .....

33. A 60 cm high cylinder with 14 cm its diameter is cut vertically into two equal halves. What is the volume of a half part?

Here,       Height (h) =14cm,       diameter (d) = 14 cm.    So,        radius (r) = 142cm                          = .....

34. The diameter of the cylindrical log of wood is 42 cm. If the area of its curved surface is 1188 cm2, find its height.

Given,           Diameter of the cylindrical log of wood (d) = 42 cm           Radius of the cylindrical log of wood (r)  = 21 cm.           Area of the curved .....

35. A solid metallic sphere of 21 cm is melted to form a solid right cylinder of radius 7 cm. Find the height of the cylinder.

Given,        Radius of the sphere (R) = 18 cm        Radius of cylinder (r) = 7 cm. Now,         Volume of sphere (V)  = 43  πR3             .....

36. The diameter of a cylinder of length 10 cm measures 14 cm. Find the curved surface area of the cylinder.

Given,        Length of the cylinder (h) = 10 cm        Diameter of the cylinder (d) = 14 cm        Radius of the cylinder (r)  = 7 cm. We know,         Surved .....

37. 35 identical circular plates each of thickness 0.8 cm and diameter 28 cm are placed one after another forming a right circular cylinder. Find the total surface area of the cylinder so formed.

Given,         Total number of circular plates (N)     = 35         Thickness of each plate (t)     = 0.8 cm So,   the height of the cylinder so formed (h) = N ⨉ .....

38.  A cylindrical water vessel of diameter 3.5 m holds 770 liters of water. Calculate the height of the vessel. (1 l = 1000 cm3) 

Given,        Diameter of water vessel (d) = 3.5 m        Radius of water vessel (r) = d2 = 1.75 m        Capacity of the vessel (V) = 770 l            .....

39. The volume of the sphere is 4851 cm3. The sphere is divided into two halves. Find the total surface area of each half.
 

Given,       Volume of the sphere (V) = 4851 cm3       Let r be the radius of sphere. We know,            Volume of the sphere (V) = 43πr3     Or, 4851 cm3  = 4 3⨉ .....

40. The length of a cylindrical wooden log is 1.5 m and its radius of base is 42 cm. What is the volume of wooden log? 

Given,          Length of wooden log (h) = 1.5m           Radius of base (r) = 42cm = 0. 42m          Volume of log = ? We know that,         Volume .....

41. The diameter of the cylindrical log wood is 28 cm. If the area of the curved surface is 7392 sq. cm, find its height.

Given,        Curved surface area = CSA = 7392cm2.         Diameter = d = 28cm. We know,         CSA = 2πrh = πdh   Or, h  = CSAπd         .....

42. The total surface area of a solid sphere made of earth is 616 cm2. Two hemispheres are formed when it is cut into two equal parts. Find the total surface area of each hemisphere.

Let r be the radius of sphere. Given,        Total surface area of the sphere (TSA) = 616 cm2 Or,  4πr2 = 616 cm2 Or,  πr2  = 154cm2. Now, Total surface area of the hemisphere (TSA) = 3πr2     .....

43. A hollow sphere made of iron has internal and external radii 5 cm and 8 cm respectively. Find the volume of iron in the sphere.

Given,         Internal radius of the hollow sphere (r) = 5 cm         External radius of the hollow sphere (R) = 8 cm. We know,         Volume of the hollow sphere (V) = 43 π (R3 .....

44. The total surface area of an iron ball is 1,810.28 cm2. If 1 cubic cm of iron costs Rs. 15, find the total cost of making the ball.

Given,          Total surface area of iron ball (TSA) = 1810.28 cm2           Rate of iron (C) = Rs. 15 per cm3 We know,        Total surface area of the iron ball (TSA) = .....

45. The circumference of the great circle of a sphere is 44 cm. Find the surface area and the volume of the sphere.

Given, Circumference of the circle (C) = 44 cm Or,  2πr = 44 cm              [Since,C = 2πr ] Or,   2 ⨉ 3.14 ⨉ r = 44 cm Or,   6.28 ⨉ r = 44 cm Or,   r = 7 cm So, the .....

46. The external and internal radii of a spherical shell of iron are 14 cm and 12 cm respectively. Find the thickness and the volume of the iron contained in the shell.

Given,         External radius of the spherical shell (R) = 14 cm         Internal radius of the spherical shell (r) = 12 cm. Then,         Thickness of the spherical shell (t) = R – .....

47. 50 equal circular plates each of diameter 15 cm and thickness 7 cm are placed one above another. Find the curved surface area and the area of its base.

Here, Given,         Total number of circular plates (N) = 50          Diameter of each plate (d) = 15 cm         Radius of each plate (r)  = 7.5 cm         .....

48. The total cost of a pipe 15 cm long is Rs. 9428.57 at the rate of Rs. 5 per cm. If the thickness of the pipe is 4 cm, find the internal radius of the pipe.

Given,         Length of the pipe (h) = 15 cm         Rate of the pipe (C) = Rs. 5 per m3         Total cost of the pipe (T) = Rs. 9428.57          Thickness of the .....

49. A pipe measures 15 cm in length. If the outer and inner radius of the pipe are 4 cm and 2 cm respectively, find the total surface area of the pipe.

Given,       Length of the pipe (h) = 15 cm       Outer radius of the pipe (r1) = 4 cm       Inner radius of the pipe (r2) = 2 cm. We know,      Total surface area of the pipe .....

50. The volume of the cylinder is 3080 cm3. If the diameter of the base is 28 m, find its depth. Also calculate the cost of painting inside the curved surface at the rate of Rs.4 per sq. m.

Given,         Volume of the cylinder (V) = 3080 cm3         Diameter of the base (d) = 28 m         Radius of the base (r) =14 m         Depth of the cylinder (h) = .....