Questions about volume and surface area with solutions for exams

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Introduction

Volume and surface area are two important concepts in geometry that help us measure the size of 3D objects. Surface area is the total area of an object's face, whereas volume is the quantity of space inside an object. Different shapes have different formulas for calculating their volume and surface area. Here in this Nithra Jobs article, we are going to see surface area and volume formulas with practice sums.

Volume and Surface area

Formulas:

CUBOID

Let length = l, breadth = b and height = h units.

Then Volume = (l * b * h) cubic units.

Surface area = 2(lb + bh + lh) sq. units.

Diagonal = √(l² + b² + h²) units.

CUBE

Let each edge of a cube be of length a.

Then, Volume = a³ cubic units.

Surface area = 6a² sq. units.

Diagonal = √3a units.

CYLINDER

Let radius of base = r and Height (or length) = h.

Then, Volume = (πr²h) cubic units.

Curved surface area = (2πrh) sq. units.

Total surface area = 2πr(h + r) sq. units.

CONE

Let radius of base = r and Height = h. Then,

Slant height, l = √(h² + r²) units.

Volume = 1/3 πr²h cubic units.

Curved surface area = (πrl) sq. units.

Total surface area = (πrl + πr²) sq. units.

SPHERE

Let the radius of the sphere be r.

Then, Volume = (4/3) πr³ cubic units.

Surface area = (4πr²) sq. units.

HEMISPHERE

Let the radius of a hemisphere be r. Then,

Volume = (2/3) πr³ cubic units.

Curved surface area = (2πr²) sq. units.

Total surface area = (3πr²) sq. units.

Solved Problems

1. 50 circular plates each of diameter 14cm and thickness 0.5cm are placed one above the other to form right circular cylinder. Find its total surface area.

Answer: 1408 cm²

Explanation:

Diameter of each circular plate = 14cm Radius = 7cm

Thickness of each plate = 0.5cm

50 plates are placed one over the other Height of the cylinder = 0.5 * 50

= 25cm

Total surface area of the cylinder = 2πr ( r + h )

= 2 * ( 22 / 7 ) * 7 ( 7 + 25 )

= 44 * 32

= 1408 cm²

Total surface area is 1408 cm²

2. The radii of the bases of a cylinder and a cone are in the ratio of 3:4 and their heights are in the ratio 2:3. Find the ratio of their volumes.

Answer: 9 : 8

Explanation:

Let r1 and h1 be the radius and height of the cylinder and r2 and h2 be the radius and height of the cone.

We have, r1 / r2 = 3 / 4 h1 / h2 = 2 / 3

Volume of the cylinder / Volume of the cone

= ( π * r1²h1 ) / ( π/3 * r2²h2 )

= 3 * (r1/r2)² * (h1/h2)

= 3 * ( 3 / 4 )² * ( 2 / 3 )

= ( 9 / 8 )

The ratio of the volume of the cylinder to the volume of the cone is 9 : 8

3. 25 circular plates each of radius 10.5cm and thickness 1.6 cm are placed one above the other to form a solid circular cylinder. Find the curved surface area and volume of the cylinder so formed.

Answer: 2640 cm² and 13860 cm².

Explanation:

Given that 25 circular plates each with radius 10.5 cm and Thickness is 1.6 cm.

Since plates are placed one above the other so its height becomes h = ( 1.6 * 25 )

h = 40 cm

Volume of the cylinder = πr²h

= ( 22 / 7 ) * 10.5 * 10.5 * 40

= ( 22 / 7 ) * 4410

= 22 * 630

= 13860 cm²

Curved surface area of a cylinder = 2πrh

= 2 ( 22 / 7 ) * 10.5 * 40

= 2 * ( 22 / 7 ) * 420

= 2 * 22 * 60

= 2640 cm²

The curved surface area and volume of the cylinder is 2640 cm² and 13860 cm².

4. The edges of cuboid are 6 cm, 7 cm and 8 cm. Find its surface area?

Answer: 292 cm²

Explanation:

From the given data,

The edges of cuboid are 6 cm, 7 cm and 8 cm Surface area of cuboid = 2 * [ lb + bh + lh ]

= 2 * [ ( 6 * 7 ) + ( 7 * 8 ) + ( 8 * 6 ) ]

= 2 * [ 42 + 56 + 48 ]

= 2 * 146

= 292 cm²

The surface area of cuboid = 292 cm².

5. The edges of a cuboid are 3 cm, 4 cm and 10 cm respectively. Find the length of the diagonal of cuboid?

Answer: 5 √5 cm

Explanation:

From the given data,

The edges of a cuboid are 3 cm, 4 cm and 10 cm

The length of the diagonal of cuboid, d = √(a² + b² + c²)

The length of the diagonal of cuboid, d = √(3² + 4² + 10²

= √( 9 + 16 + 100 )

= √( 125 )

= √( 25 * 5 )

= 5 √5

The length of the diagonal of cuboid = 5 √5 cm.

Conclusion

By learning about volume and surface area, we can understand how to compare and contrast different 3D objects based on their size and shape. We can also apply these concepts to real-world problems such as finding the capacity of a container, the amount of material needed to cover a surface, or the efficiency of a design. Volume and surface area are useful tools for exploring the geometry of our world.