**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – 3D-Mensuration**

Contents

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Every real object occupies some space. It is usually specified by its three dimensions-Length, Breadth and Depth (or height or thickness). The object may be solid or hollow, in case of circular, cylindrical and spherical object, the specifying . dimensions may change to radius, angle etc.

**Features of 3D-Mensuration**

**Volume** The amount of space occupied by the 3D-object is called its volume. It is measured in m³, cm³.

**Surface Area** The sum of area of at all the surfaces of the object is called surface area. It is measured in m², cm².

**Properties and Formulae of Important 3D-Figures**

**Cube** A solid body having 6 equal faces with equal length, breadth and height is called a cube. In fact, each face of a cube is a square.

**Cuboid** A cuboid is made up of 6 rectangular faces, equal.

**Example The length (in m) of the longest rod that can be put in a room of dimensions 10m x 10m x 5m is SSC (10 + 2) 2010**

(a) 15√3

(b) 15

(c)10√2

(d)5√3

**Room** A rectangular room has 4 walls (surfaces) and opposite walls have equal areas.

• Total area of walls = 2 (l+ b)xh

• Total volume of the room = lbh

• Area of floor or roof = l x b

Where, l = Length, b = Breadth, h = Height

**Example A hall is 15 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas of the four walls, the volume of the hall, in cubic metres, is ssc (10+2) 2011**

(a) 720

(b) 900

(c) 1200

(d) 1800

**Right Circular Cylinder** It is formed by keeping circles of equal radii one on another. Also, it is formed by rolling a rectangular sheet. It can be of two types.

**1. Solid Right Circular Cylinder**

**Right Circular Cone** A right circular cone is a solid obtained by rotating a right angled triangle around its height.

**Example The radius of the base of a right circular cone is doubled. To keep the volume fixed, the height of the cone will be ssc (1o+2) 2012**

(a) half of the previous height

(b) one-third of the previous height

(c) one-fourth of the previous height

(d) 1/√2 times of the previous height

**Sphere** It is made up of all points in the space, which lie at a constant distance from a fixed point. That constant distance and fixed point are respectively called the radius and center of the sphere.

• Volume of the sphere = 4/3πr³

• Total surface area = 4πr² ; where, r = Radius

**Hollow Sphere** Its both external and internal surface are spherical and both the surfaces have a common central point.

• Volume of the hollow sphere=4/3π(R³-r³)

• Internal surface area = 4πr²

• External surface area = 4πR²

Where, R= External radius, r= Internal radius

**Hemisphere** It is the half part of a sphere.

• Volume of the hemisphere = 2/3πr³

• Total surface area = 3πr²

• Curved surface area = 2πr², where r = Radius

**Right Prism**

• Volume of the prism = Area of base x Height

• Lateral surface area of prism = Perimeter of base x Height

• Total surface area of prism

= Lateral surface area of prism + 2 x Area of base

**Regular Right Pyramid with Triangular or Square Base**

There are two types of pyramid

• Volume of pyramid

= 1/3 x Area of base x Height

• Total surface area of pyramid = Area of base + Area of slant surface

For triangular pyramid Base = abc,

Slant surface = abd + bcd + acd

For square pyramid Base = abcd,

Slant surface = apd + dpc + pcb + apb

**Essential Points**

**Reference Corner**

**1.A rectangular swimming pool of length 25 m and breadth 12m has a uniformly sloping floor. It is 2 m in depth on one end of its length and 3 m at the other. How much water is needed to completely fill this pool? SSC (10 + 2) 2017**

(a) 1500 cu m

(b) 2250 cu m

(c) 3000 cu m

(d) 750 cu m

**Answer:**

(d)

**2.The radius of a hemispherical bowl is 6 cm. The capacity of the bowl is SSC (10 + 2) 2015**

(a) 495.51 cm³

(b) 452.57 cm³

(c) 345.53 cm³

(d) 452 cm³

**Answer:**

**3.If the radius of a sphere be doubled, then the percentage of increase in volume is SSC (10+2) 2014**

(a) 600%

(b) 800%

(c) 500%

(d) 700%

**Answer:**

**4. Three solid iron cubes of edges 4 cm, 5 cm and 6 cm are melted together to make a new cube. 62 cm³ of the melted material is lost due to improper handling. The area (in cm²) of the whole surface of the newly formed cube is SSC (10+2) 2013**

(a) 125

(b)216

(c) 294

(d) 343

**Answer:**

**5.If the ratio of volumes of two cones is 2 : 3 and the ratio of the radii of their bases is 1 : 2, then the ratio of their heights will be SSC (10 + 2) 2013**

(a) 3 : 8

(b) 8 : 3

(c) 3 : 4

(d) 4 : 3

**Answer:**

**6. A right pyramid stands on a base 16 cm square and its height is 15 cm. The area (in cm²) of its slant surface is SSC (10 + 2) 2013**

(a) 344

(b) 444

(c) 514

(d) 544

**Answer:**

**7.A solid iron sphere of diameter 6 cm is melted and made to a hollow cylindrical pipe of external diameter 10 cm and of length 4 cm. The width of the sheet of pipe is SSC (10 +2) 2013**

(a) 1 cm

(b) 2 cm

(c) 3 cm

(d) 4 cm

**Answer:**

**8. If the volumes of a sphere and a cylinder of equal radius are same, then ratio of the radius and height of the cylinder will be SSC (10 +2) 2013**

(a) 4:3

(b)3:2

(c) 3 : 4

(d) 2 : 3

**Answer:**

**Practice Exercise**

**1. In a swimming pool measuring 90 m by 40 m, 150 men take a dip. If the average displacement of water by a man is 8 m³, what will be the rise in water level?**

(a) 30 cm

(b) 33.33 cm

(c) 20 cm

(d) 25 cm

**2. A hemispherical basin of 150 cm diameter holds water one hundred and 20 times as much as a cylindrical tube. It the height of the tube is 15 cm, then the diameter of the tube is**

(a) 27 cm

(b) 24 cm

(c) 25 cm

(d) 26 cm

**3. A cylindrical box of radius 5 cm contains 10 solid spherical balls each of radius 5 cm. If the topmost ball touches the upper cover of the box, then volume of the empty space in the box is**

(a) 2500/3 π cm³

(b) 5000 π cm³

(c) 2500 π cm³

(d) 5000/3 π cm³

**4.Water flows into a tank 100 m by 150 m through a rectangular pipe 1.5 m x 1.25 m at a speed of 20 km/h. In what time will the water rise by 2 m?**

(a) 96 min

(b) 50 min

(c) 48 min

(d) 75 min

**5. How many metres of cloth 10 m wide will be required to make a conical tent with base radius of 14 m and height is 48 m?**

(a) 110 m

(b) 55 m

(c) 77 m

(d) 220 m

**6. A copper sphere of diameter 36 cm is drawn into a wire of diameter 4 mm. Find the length of the wire, **

(a) 2000 m

(b) 1500 m

(c) 1855 m

(d) 1944 m

**7. A well with 28 m inside diameter is dug 8 m deep. The earth taken out of it has been evenly spread around it to a depth of 21 m to form an embankment. Find the height of the embankment.**

(a) 215.4 m

(b) 218.8 m

(c) 213.3 m

(d) 220.7 m

**8. A cone has height half of 16.8 cm while diameter of its base is 4.2 cm. It is melted and recast into a. sphere. Find the surface area of the sphere.**

(a) 55.44 sq cm

(b) 60 sq cm

(c) 65.58 sq cm

(d) 59 sq cm

**9. Find the area of the iron sheet required to prepare a cone double the height of 12 cm with diameter of the base 14 cm**

(a) 504 sq cm

(b) 804 sq cm

(c) 304 sq cm

(d) 704 sq cm

**10. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 10 per.**

(a) Rs 50

(b) Rs 6

(c) Rs 98

(d) Rs 55

**11. A hemispherical bowl has 3.5 cm radius. It is to be painted inside as well as outside. Find the cost of painting it at the rate of Rs 5 per 10 sq cm.**

(a) Rs 50

(b) Rs 81

(c) Rs 56

(d) Rs 77

**12. A hollow cylindrical tube 20 cm long is made of iron and its external arid internal diameters are 8 cm and 6 cm, respectively. The volume of iron used in making the tube is**

(a) 1764 cu cm

(b) 880 cu cm

(c) 440 cu cm

(d) 220 cu cm

**13.A hollow iron pipe is 21cm long and its exterior diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weights 8 g/cm³, then the weight of the pipe**

(a) 3.696 kg

(b) 3.6 kg

(c)36kg

(d) 36.9 kg

**14.A solid metal ball of diameter 16 cm is melted and cast into smaller balls, each of radius 1 cm. The number of such balls is**

(a) 256

(b) 2048

(c) 512

(d) 4096

**15.If base of a prism is a square of side 4 cm. If the height of prism is 10 cm, then what will be total surface area of that prism **

(a) 192 cm²

(b) 212 cm²

(c) 214 cm²

(d) None of these

**16. A triangular base pyramid has are 25 cm² and height 14 cm, then what is the volume of that pyramid?**

(a) 146 cm³

(b) 175 cm³

(c) 178 cm³

(d) None of these

**17. The volume of a pyramid of base area 25 cm² and height 12 cm is**

(a) 200 cm³

(b) 100 cm³

(c) 400 cm³

(d) 800 cm³

**18. If the base of right rectangular prism remains constant and the measures of the lateral edges are halved, then its volume will be reduced by**

(a) 50%

(c) 66.60%

(b) 33.33%

(d) None of these

**Answers**

**Hints & Solutions**

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