# Volume & Surface Areas

Volume and surface area are related to solids or hollow bodies. These bodies occupy space and have usually three dimensions length, breadth and height.

## Volume

Space occupied by an object is called the 'volume' of that particular object. It is always measured in cube unit like cubic meter, cubic centimetre etc.

*For example* If length, breadth and height of box are 9 cm, 4 cm and 2 cm respectively, then Volume of the box

= Length x breadth x Height

= 9 x 4 x 2 = 72 cm^{3}

## Surface Area

Surface area of a solid body is the area of all of its surfaces together. Surface area is measured in square unit like square meter, square centimetre etc.

*For example:* A cube has 6 surfaces and each surface is in a square like shape. Therefore, its surface area will be 6a^{2} sq units, where a^{2} is the area of each surface of the cube

__Cracking Aptitude Questions on Volume and Surface Area__

__Cracking Aptitude Questions on Volume and Surface Area____Part – I: __Cube

Let the length of each edge of a cube be ‘a’. Then,

1. Volume = a^{3} cubic units.

2. Surface area = 6a^{2} sq. units.

3. Diagonal = a θ3 units.

**Question: **A large cube is formed from the material obtained by melting three smaller cubes of 3, 4 and 5 cm side. What is the ratio of the total surface areas of the smaller cubes and the large cube?

**Solution:**

Volume of the larger cube = Total volume of the smaller cubes

= (3^{3} + 4^{3} + 5^{3}) cm^{3} = 216 cm^{3}

Side of the cube = (216)^{1/3} = 6 cm

Surface area of the larger cube = 6 x 6 x 6 = 6 x 36 cm^{2}

Total surface area of the smaller cubes = 6 x (3^{2} + 4^{2} + 5^{2}) cm^{2} = 6 x 50 cm^{2}

**Required ratio = 50 / 36 = 25:18.**

**Question: **A 4cm cube is cut into 1 cm cubes. What is the total surface area of the new cubes?

**Solution:**

Number of cubes that can be formed = Total volume / Volume of each cube

= (4 x 4 x 4) / (1 x 1 x 1) = 64.

Total surface area of all cubes = Number of cubes x Surface area of 1 cube = 64 x (6 x 1 x 1) = **384 cm**^{2}

__Part – II:__ Cuboid

Let the length, breadth and height be ‘l’, ‘b’ and ‘h’ respectively. Then,

1. Volume = (l x b x h) cubic units.

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

3. Diagonal = ? (l^{2} + b^{2} + h^{2)} units.

**Question: **50 men took a dip in a water tank 40 m long and 20 m broad on a religious day. If the average displacement of water by a man is 4 m^{3}, then calculate the rise in the water level.

**Solution:**

Total volume of water displaced = 50 x 4 m^{3} = 200 m^{3}

Change in volume of the tank occupied = length x breadth x change in height.

=> ** Change in height = 200 m**^{3}** / (40 m x 20 m ) = 0.25 m = 25 cm.**

**Question: **A cistern 6m long and 4 m wide contains water up to a depth of 1 m 25 cm. What is the surface area of the wet surface?

**Solution:**

Surface area of wet surface = Area occupied – Area of top surface (since it is an open surface)

= Area of cuboid – Area of base

= 2(lb + bh + lh) – lb

= 2(6x4 + 4x1.25 + 6x1.25) – 6x4

= 2x1.25x10 + 24 = **49 m**^{2}

__Part – III__: Cylinder

Let radius of the base of a cylinder be ‘r’ and its height be h. Then,

1. Volume = ? (r^{2}h) cubic units.

2. Curved surface area = (2 ? rh) sq. units.

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

**Question: **A hollow iron pipe is 21 cm long and its external diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs 8 g/cm^{3}, what is the weight of the pipe?

**Solution:**

External radius R = 4 cm, Internal radius r = (4 – 1) cm = 3 cm

Volume of the pipe = Total solid volume – Volume of hollow part

= ?(R^{2} – r^{2})h

= 22 / 7 x (4x4 – 3x3) x 21 = 462 cm^{3}.

Mass of the pipe = density x volume

**= 8 x 462 = 3696 gm = 3.696 kg.**

**Question: **66 cm^{3} of silver is drawn into a wire 1 mm in diameter. Find the length of the wire in meters.

**Solution: **

Radius of the wire r = 0.5 mm = 0.05 cm and let length = l.

Volume of the wire = 66 cm^{3} = ? r^{2} l = 22 / 7 x 0.05 x 0.05 x l

=> ** l = 66 x 7 / (22 x 0.05 x 0.05) = 8400 cm = 84 m.**

__Part – IV__: Sphere and Hemi-Sphere

Let the radius of the sphere be r. Then,

1. Volume = (4 / 3) ? r^{3} cubic units.

2. Surface area = (4 ? r^{2}) sq. units.

For a hemisphere of radius r, we have

1. Volume = (2 / 3) ? r^{3} cubic units.

2. Curved surface area = (2 ? r^{2}) sq. units.

3. Total surface area = (3 ? r^{2}) sq. units.

**Question: **The volume of a spherical ball is 121000 cm^{3}. What is the radius of the ball?

**Solution:**

(4 / 3) ? r^{3} = 121000

=> r^{3} = 121000 x 3 / 4 x 7 / 22 = 28875

=> **r = (28875)**^{1/3}** = 30.68 cm.**

**Question: **Calvin is making a model Earth in his science class. The radius of the inner core of the model Earth is 4 cm and the radius of the entire model is 10 cm. How many times larger is the volume of the entire model than the volume of the inner core?

**Solution: **

Volume of core = (4 / 3) ? x 4 x 4 x 4 cm^{3}

Volume of earth = (4 / 3) ? x 10 x 10 x 10 cm^{3}

Volume of entire model = (1000 / 64) times the volume of the core

= **15.625 times the volume of the core.**

__Part – V:__ Cone

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

1. Slant height, l = ?(h^{2} + r^{2}) units.

2. Volume = (? r^{2}h / 3) cubic units.

3. Curved surface area = (? rl) sq. units.

4. Total surface area = ?rl + ? r^{2} sq. units.

**Question:** A right triangle with sides 3 cm, 4 cm and 5 cm is rotated along the side of 3 cm to form a cone. Find the volume of the cone so formed.

**Solution:**

From the question, we can say that r = 3 cm, h = 4 cm and l = 5 cm.

Thus, **volume of the cone = ? x 3 x 3 x 4 / 3 **

** = 12 ? cm**^{2}**.**

**Question:** The slant height of a right circular cone is 10 m and its height is 8 m. Find the area of its curved surface.

**Solution: **

Given, l = 10 m and h = 8 m.

r = ?(l^{2} – r^{2}) = ?(100 – 64) = 6 m

**Curved surface area = ? x r x l **

** = ? x 6 x 10 = 60 ? m**^{2}**.**