Mensuration- II (Area & Volume)

 Solid Geometry 

Rectangular Solid / Cuboid

A rectangular solid or box is a solid formed by six rectangles called faces.

The sides of the rectangles are called edges. 

The edges are called length(l), breadth (w) and height (h).

• Volume = V = l × w × h
• Area = A = 2 (lw + wh + hl)

A box has 4 diagonals, all the same length. 

A diagonal of a box is the longest line segment that can be drawn between two points of box.

So, diagonal  = d = √(l² + w² + h²)

Cube

A cube is a rectangular solid in which length, breadth and height are equal. So, all the edges are of same length. 

• Volume = V = e³ ( e - edge of Cube)
• Area = A = 6e²
• Body diagonal or diagonal = e√3
• Face diagonal = e√2
• Lateral Surface Area = 4e²
• Area of 4 walls of a room = 2 × h ×(l + w)

Cylinder 

• Volume V of a Cylinder whose circular base has radius r and height h is V = πr²h
• Curved Surface Area = Lateral Surface Area = 2πrh
• Area of top and bottom part = 2πr²
• Total Surface Area = 2πrh + 2πr² = 2πr(h + r)

Spehere

• Volume = V = 4/3 πr³
• Surface Area = Curved Surface Area = A = 4πr²

          where r = radius 

Semi-spehere 

• Volume = V = 2/3 πr³
• Curved Surface Area = 2πr²
• Total Surface Area = A = 2πr² + πr² = 3πr²

Cone

• Slant Height = l = √(r² + h²)
• Volume = V = 1/3 πr²h
• Curved Surface Area = πrl
• Total Surface Area = πrl + πr² = πr(l + r)

Where, r = radius & h = height 

Spherical Cell

• Volume = 4/3 π(R³ - r³)
• Total Surface Area = 4π(R² - r²)

Where, R - Outer radius 
              r - Inner radius 

Examples 

1. How many spherical bullets can made out of a lead cylinder 28 cm high and with base radius 6 cm, each bullet being 1.5 cm in diameter? (Ans. 1792)

2. A copper sphere of diameter 18 cm is drawn into a wire of diameter 4 mm. Find the length of the wire. (Ans. 243 m)

3. A hemispherical bowl of internal radius 9 cm contains a liquid.  This liquid is to be filled into cylindrical shaped small bottles of diameter 3 cm and height 4 cm. How many bottles will be needed to empty the bowl ? ( Ans. 54)

4. Find the length of canvas 1.25 m wide required to build a conical tent of base radius 7 m and height 24 m. (Ans. 440 m)

5. The volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is 12.8 cu. metres. Find the breadth of the wall. ( Ans. 40 cm)

Previous Years Questions 

1. How many small cubical blocks of side 6 cm can be cut from a cubical block whose each side measures 24 cm ? (Ans. 64)

2. The curved surface area of a cylindrical pillar is 264 sq.m. and its volume is 924 cu.m. The ratio of of its diameter to height is :----- (Ans. 7 : 3)

3. A spherical lead ball of radius 10 cm is melted and small lead balls of radius 5 mm are made. The total numbers of possible small lead balls is :---? (Ans. 8000)

4.  A solid cylinder has total surface area of 462 sq. cm. Curved surface area is 1/3rd of its total surface area. The volume of the cylinder is

A. 530 cm³ 

B. 536 cm³ 

C. 539 cm³ 

D. 545 cm³ 

5. A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8:5, the ratio of their radius and height is 

A. 1:2

B. 2:3

C. 1:3

D. 3:4

6. The radii of the base of a cylinder and a cone are in the ratio √3: √2 and their heights are in the ratio √2: √3. Their volumes are in the ratio of

A. √3:√2 

B. 3√3:√2

C. √3:2√2

D. √2: √6

7. The radius of the base of a right circular cone is doubled keeping its height fixed. The volume of the cone will be:

A. three times of the previous volume

B. four times of the previous volume

C. √2 times of the previous volume 

D. double of the previous volume

8. The diameter of a cylinder is 7 cm and its height is 16 cm. The lateral surface area of the cylinder is

A. 352 cm²

B. 350 cm²

C. 355 cm²

D. 348 cm²