If ' V ' is the volume of a cuboid of dimensions a × b × c and ' S ' is its surface area, then prove that 1 V = 2 S 1 a + 1 b + 1 c .

Given: ' V ' is the Volume of the cuboid, ' S ' is the Surface area of the cuboid and a × b × c are the dimensions of the cuboid. To prove: 1 V = 2 S 1 a + 1 b + 1 c . We know that, S = 2 a b + b c + c a and V = a b c 1 V = 1 a b c Multiply the numerator and denominator by S . 1 V = S S a b c 1 V = 2 a b + b c + c a S a b c {Since, S = 2 a b + b c + c a } 1 V = 2 S a b a b c + b c a b c + c a a b c 1 V = 2 S 1 c + 1 a + 1 b Re-arranging the terms, 1 V = 2 S 1 a + 1 b + 1 c . Hence, proved.

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The volume of a cube is $512$ cubic metre. Find its edge in $m$ .

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Important Questions on Surface Areas and Volumes

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

You have a

9 \times 9 \times 9 cm

box. How many cubes of side

5 cm

will you be able to fit inside it?

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Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

A platform with dimensions

22 m \times 14 m \times 2.5 m

is formed from the earth digging the well with diameter

7 m

. Find the depth of the well.

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Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

Each side of a cube is decreased by

50 % .

Calculate the ratio of the volumes of the original and changed cube.

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Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

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 $cu m$ is

HARD

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

On each face of a cuboid, the sum of its perimeter and its area is written. Among the six numbers so written, there are three distinct numbers and they are

16, 24

and

31

. The volume of the cuboid lies between

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

Some bricks are arranged in a pile with volume

30 cu m

. If the length, breadth and height of each brick is

25 cm

12.5 cm

and

8 cm

respectively, then in that pile the number of bricks are (Assume there is no gap in between two bricks)

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

The volume of the cuboid, whose length, breadth and height are

12 m, 10 m and 8 m

respectively is:

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

If total surface area of a cube is

96 {cm}^{2}

, then its volume is

MEDIUM

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

The length of a cuboid is twice its breadth and its height is half of its breadth. If the height of the cuboid is

2 cm

, then what is the edge of a cube having the same volume as that of the cuboid?

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

The volume of a cube is $1000 {cm}^{3}$ . Find its side.

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

The edge of a cube is

5 cm

. What will be its volume?

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

Two cones have their heights in the ratio of

1 : 3

and radii in the ratio

3 : 1

. What is the ratio of their volumes?

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

The number of cubes of side

2 cm

which can be cut from a cube of side

6 cm

HARD

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

64

buckets of water are withdrawn from a cubical water lank, full of water, then

\frac{1}{3}

of water in the tank still remains. If the length of the side of the water tank is

12

metre, then what is the capacity (in litre) of each bucket?

(1 cubic decimeter = 1 litre)

MEDIUM

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

V

is the volume of cuboid whose length is

a

, breadth is

b

and height is

c

and

S

is its total surface area, then prove that

\frac{1}{V} = \frac{2}{S} [\frac{1}{a} + \frac{1}{b} + \frac{1}{c}]

MEDIUM

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

Sum of the length of all edges of a cube is

x

metres. If the surface area of the cube is

x sq

. metres, then its volume (in cubic metres)

EASY

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

From a rectangular sheet of dimensions

32 cm \times 18 cm

, a square of side

3 cm

is cut from the four corners of the sheet and a box is made. The volume of the box is:

HARD

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

If ' $V$ ' is the volume of a cuboid of dimensions $a \times b \times c$ and ' $S$ ' is its surface area, then prove that $\frac{1}{V} = \frac{2}{S} (\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ .

MEDIUM

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

A cone of base radius

4 cm

is divided into two parts by drawing a plane through the mid-point of its height and parallel to its base. Compare the volume of the two parts.

MEDIUM

Mathematics>Geometry>Surface Areas and Volumes>Volume of a Combination of Solids

Water in a canal,

6 m

wide and

1.5 m

deep, is flowing with a speed of

10 km / hour

. How much area will it irrigate in

30

minutes; if

8 cm

standing water is needed? (write answer in

m^{2}

)

The volume of a cube is 512 cubic metre. Find its edge in m.

Important Questions on Surface Areas and Volumes

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The volume of a cube is $512$ cubic metre. Find its edge in $m$ .