# Volume and Surface Area Questions

FACTS  AND  FORMULAE  FOR  VOLUME  AND  SURFACE  AREA  QUESTIONS

I. CUBOID

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

1. Volume = (l x b x h)

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

3. Diagonal =$\sqrt{{l}^{2}+{b}^{2}+{h}^{2}}$ units

II. CUBE

Let each edge of a cube be of length a. Then,

1. Volume = ${a}^{3}$ cubic units.

2. Surface area = $6{a}^{2}$ sq.units

3. Diagonal = $\sqrt{3}a$ units

III. CYLINDER

Let radius of base = r and Height (or Length) = h. Then,

1.Volume = $\left({\mathrm{\pi r}}^{2}\mathrm{h}\right)$ cubic units

2. Curved surface area = $\left(2\mathrm{\pi }rh\right)$ sq.units

3. Total surface area = $\left(2\mathrm{\pi rh}+2{\mathrm{\pi r}}^{2}\right)$ sq.units

IV. CONE

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

1. Slant height, $l=\sqrt{{h}^{2}+{r}^{2}}$ units

2. Volume = $\left(\frac{1}{3}{\mathrm{\pi r}}^{2}\mathrm{h}\right)$ cubic units.

3. Curved surface area = $\left(\mathrm{\pi rl}\right)$sq.units

4. Total surface area = $\left(\mathrm{\pi rl}+{\mathrm{\pi r}}^{2}\right)$sq.units

V. SPHERE

Let the radius of the sphere be r. Then,

1. Volume =$\left(\frac{4}{3}{\mathrm{\pi r}}^{3}\right)$ cubic units

2. Surface area = $\left(4{\mathrm{\pi r}}^{2}\right)$ sq.units

VI. HEMISPHERE

Let the radius of a hemisphere be r. Then,

1. Volume = $\left(\frac{2}{3}{\mathrm{\pi r}}^{3}\right)$ cubic units.

2. Curved surface area = $\left(2{\mathrm{\pi r}}^{2}\right)$ sq.units

3. Total surface area = $\left(3{\mathrm{\pi r}}^{2}\right)$ sq.units

Q:

If each edge of a cube is increased by 50%, find the percentage increase in Its surface area

 A) 125% B) 150% C) 175% D) 110%

Explanation:

Let the edge = a cm

So increase by 50 % = a + a/2 = 3a/2

Total surface Area of original cube = $6{a}^{2}$

TSA of new cube = $6{\left(\frac{3a}{2}\right)}^{2}$ =$6\left(\frac{9{a}^{2}}{4}\right)$=  $13.5{a}^{2}$

Increase in area = $13.5{a}^{2}-6{a}^{2}$ =$7.5{a}^{2}$

$7.5{a}^{2}$ Increase % =$\frac{7.5{a}^{2}}{6{a}^{2}}×100$ = 125%

296 55208
Q:

How many cubes of 3cm edge can be cut out of a cube of 18cm edge

 A) 36 B) 232 C) 216 D) 484

Explanation:

number of cubes=(18 x 18 x 18) / (3 x 3 x 3) = 216

71 21825
Q:

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 four walls, the volume of the hall is:

 A) 720 B) 900 C) 1200 D) 1800

Explanation:

40 20137
Q:

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

 A) 12 pi cub.cm B) 15 pi cub.cm C) 16 pi cub.cm D) 20 pi cub.cm

Answer & Explanation Answer: A) 12 pi cub.cm

Explanation:

42 19708
Q:

A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes. Find the least possible number of cubes.

 A) 30 B) 40 C) 10 D) 20

Explanation:

Volume of the block = (6 x 12 x 15) cu.cm = 1080 cu.cm

Side of the largest cube = H.C.F. of 6 cm, 12 cm, 15 cm

= 3 cm.

Volume of this cube  = (3 x 3 x 3) cu.cm = 27 cu.cm

Number of cubes =$\frac{1080}{27}$ = 40.

39 18360
Q:

Three solid cubes of sides 1 cm, 6 cm and 8 cm are melted to form a new cube. Find the surface area of the cube so formed

 A) 486 B) 586 C) 686 D) 786

Explanation:

Volume of new cube =  $\left({1}^{3}+{6}^{3}+{8}^{3}\right)c{m}^{3}$$729c{m}^{3}$

Edge of new cube = $\sqrt[3]{729}$ = 9cm

Surface area of the new cube = ( 6 x 9 x 9) sq.cm = 486 sq.cm

26 14680
Q:

The diagonal of a rectangle is sqrt(41) cm.  and its area is 20 sq. cm. The perimeter of the rectangle must be:

 A) 9 cm B) 18 cm C) 20 cm D) 41 cm

Answer & Explanation Answer: B) 18 cm

Explanation:

${l}^{2}+{b}^{2}$ = ${\left(diagonal\right)}^{2}$=40

Also, lb=20

${\left(l+b\right)}^{2}={l}^{2}+{b}^{2}+2lb$ = 41 + 40 =81

(l + b) = 9.

Perimeter = 2(l + b) = 18 cm.

11 14029
Q:

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

 A) 49 B) 50 C) 53.5 D) 55

Explanation:

Area of the wet surface = [2(lb + bh + lh) - lb]

= 2(bh + lh) + lb

= [2 (4 x 1.25 + 6 x 1.25) + 6 x 4]${m}^{2}$

= 49${m}^{2}$