Cube
1. volume of cuboid = l x h x b
where, l is the length,
h is the height,
and b is the breadth of the cuboid
2. Total surface area = 2 ( l x h + l x b + b x h)
where, l is the length,
h is the height,
and b is the breadth of the cuboid
3. Relation between the diagonal of a cube and the side of a cube is
Diagonal of a cube = √a² + b² + c²
Solved Examples
1. A bricks measures 20 cm by 10 cm by 7(1/2) cm. How many bricks will be required for a wall 25 m long, 2 m high and 3/4 m thick?
Answer:
Volume of wall == 25 x 2 x (3/4) = 75/2 cubic meter= 37.5 cu meter
volume of a brick = 20 x 10 x 15/2 = 1500 cu cm = 1500/(100x100x100) cu meter = .0015 cu meter
Therefore number of bricks required = 37.5/0.0015 =25000 bricks
2.The perimeter of one face of a cube is 20 cm. Its volume must be
1. 8000 cubic cm 2. 1000 cubic cm
3. 125 cubic cm 4. 400 cubic cm
Answer: 3. 125 cubic cm
Perimeter of square = 4 x l
=> 20 = 4L
=> L = 5 cm
Therefore, volume = L x L x L = 5 x 5 x 5 = 125 cu cm
3. The length of a longest rod that can be placed in a room which is 12 m, 9 m and 8 m high, is
1. 27 m 2. 19 m
3. 17 m 4. 13 m
Answer:
3. 17 m
The length of the diagonal of the cube is the length of the longest rod that can be placed in the room.
Therefore the diagonal of the cube = √ a² +b² + c² where a , b and c are the three sides of the triangle.
Thus diagonal = √ 12² +9² + 8² = √ 144 + 81+ 64 = √ 289 = 17
4. If diagonal of a cube is √12 cm, then its volume (in cu cm) is
1. 8 2. 12
3. 24 4. 3√2
Answer: 1. 8
The diagonal of a cube is √3a where a is the side of the cube.
Thus, diagonal
=>√3 a = √12
=> √3 a = √4 x 3
=> √3 a = 2 √3
=> a = 2
And volume = 2 x 2 x2 = 8 cu cm
5. How many cubes,each of edge 3 cm, can be cut from a cube of edge 15cm?
1. 25 2. 27
3. 125 4. 144
Answer: 3. 125
From any one face of the cube we can 5 x 5 = 25 identical squares.
And the entire cube can be divided in to 15/3 = 5 layers of 25 identical cubes each.
Therefore number of cubes such formed are 25 x 5 = 125 cubes
6. A river 3 m deep and 60 m wide is flowing at the rate of 2.4 km/h. The amount of water running into the sea per minute is
1. 6000 cu m 2. 6400 cu m
3. 6800 cu m 4. 7200 cu m
Answer:4. 7200 cu m
First we convert the rate in m/s from 2.4 km/hr.
2.4 km/hr = 2.4 x 1000/60 m/min = 40 m/s
Therefore, in 1 minutes 40 meters length of the water is dumped into the sea. The cube thus formed has dimensions 40 x 60x 3 = 7200 cu meter
7. If the volume of two cubes are in ratio 27:64, then the ratio of their total surface area is
1. 27:64 2. 3:4
3. 9:16 4. 3:18
Answer: 2. 3:4
Volume of cube 1 / Volume of cube2 = 27/64
=> length of cube1 / length of cube 2 = v27/64 = 3/4
The formula for the total surface area of a cube is 6 L
Therefore, the ratio of total surface area = 6l/6L = 6 (length of cube 1)/6(length of cube 2)
= (length of cube 1)/(length of cube 2)
= 3/4
8. The height of a wall is six times its width and the length of the wall is seven times its height. If volume of the wall be 16128 cu m. Its width is
1. 5m 2. 4m
3. 4.5 m 4. 6 m
Answer:2. 4m
Let the length of the wall be L and width W and height H, then
Given that ,
H = 6W
and L = 7H or
L = 7 x 6w = 42W
Again, the volume of a cube = H x L X W
16128 = 6W x 42W x W
=> W³ = (16128 )/(6 x 42)
W³ = 64
W = 4
Therefore width is 4 cm.
9. The volume of a cuboid is twice the volume of a cube. If the dimensions of the cuboid are 9 cm,8 cm and 6 cm. The total surface area of the cube is
1. 72 sq cm 2. 216 sq cm
3. 432 sq cm 4. 108 sq cm
Answer: 6 cm
Given that the volume of cuboid is twice that of a cube
volume of cuboid = 9 x 8 x 6 = 432
Thus the volume of cube is 432/2 = 216
And the length of the cube = ∛216 = 6 cm
And thus total surface area of cube = 6 x 6 x6 = 216 sq cm
10. The edges of a cuboid are in ratio 1:2:3 and its surface area is 88 cm² .
the volume of the cuboid is
1. 120 cm³ 2. 64 cm³
3. 48 cm³ 4. 24 cm³
Answer:3. 48 cm³
Let the edges of cube be L , 2L , 3L
And the surface area is 2(L x 2L) + 2(L + 3L) + 2(2L + 3L) = 88
4L² + 8L² + 10L² = 22L² = 88
=> L² = 4
=> L = 2 cm
And the volume is 2 x 2(2) x 3(2) = 2 x 4 x 6 = 48 cm³
A Cube has 6 equal square faces. A good example is a box with equal height, length and breadth.
1. Volume of cube = L x L x L ,
where l is the length of any side.
2. Total surface area of cube = 6 X L
3. Relation between the diagonal of a cube and the side of a cube is
Diagonal of a cube = √3 x L
3. Relation between the diagonal of a cube and the side of a cube is
Diagonal of a cube = √3 x L
Cuboid
 |
| Cuboid with length l , height h, and breadth b |
A
Cuboid has 6 faces just like cube but they are not all equal. For a
cuboid at least one of the length, breadth or height is different .
1. volume of cuboid = l x h x b
where, l is the length,
h is the height,
and b is the breadth of the cuboid
2. Total surface area = 2 ( l x h + l x b + b x h)
where, l is the length,
h is the height,
and b is the breadth of the cuboid
3. Relation between the diagonal of a cube and the side of a cube is
Diagonal of a cube = √a² + b² + c²
Solved Examples
1. A bricks measures 20 cm by 10 cm by 7(1/2) cm. How many bricks will be required for a wall 25 m long, 2 m high and 3/4 m thick?
Answer:
Volume of wall == 25 x 2 x (3/4) = 75/2 cubic meter= 37.5 cu meter
volume of a brick = 20 x 10 x 15/2 = 1500 cu cm = 1500/(100x100x100) cu meter = .0015 cu meter
Therefore number of bricks required = 37.5/0.0015 =25000 bricks
2.The perimeter of one face of a cube is 20 cm. Its volume must be
1. 8000 cubic cm 2. 1000 cubic cm
3. 125 cubic cm 4. 400 cubic cm
Answer: 3. 125 cubic cm
Perimeter of square = 4 x l
=> 20 = 4L
=> L = 5 cm
Therefore, volume = L x L x L = 5 x 5 x 5 = 125 cu cm
3. The length of a longest rod that can be placed in a room which is 12 m, 9 m and 8 m high, is
1. 27 m 2. 19 m
3. 17 m 4. 13 m
Answer:
3. 17 m
The length of the diagonal of the cube is the length of the longest rod that can be placed in the room.
Therefore the diagonal of the cube = √ a² +b² + c² where a , b and c are the three sides of the triangle.
Thus diagonal = √ 12² +9² + 8² = √ 144 + 81+ 64 = √ 289 = 17
4. If diagonal of a cube is √12 cm, then its volume (in cu cm) is
1. 8 2. 12
3. 24 4. 3√2
Answer: 1. 8
The diagonal of a cube is √3a where a is the side of the cube.
Thus, diagonal
=>√3 a = √12
=> √3 a = √4 x 3
=> √3 a = 2 √3
=> a = 2
And volume = 2 x 2 x2 = 8 cu cm
5. How many cubes,each of edge 3 cm, can be cut from a cube of edge 15cm?
1. 25 2. 27
3. 125 4. 144
Answer: 3. 125
From any one face of the cube we can 5 x 5 = 25 identical squares.
And the entire cube can be divided in to 15/3 = 5 layers of 25 identical cubes each.
Therefore number of cubes such formed are 25 x 5 = 125 cubes
6. A river 3 m deep and 60 m wide is flowing at the rate of 2.4 km/h. The amount of water running into the sea per minute is
1. 6000 cu m 2. 6400 cu m
3. 6800 cu m 4. 7200 cu m
Answer:4. 7200 cu m
First we convert the rate in m/s from 2.4 km/hr.
2.4 km/hr = 2.4 x 1000/60 m/min = 40 m/s
Therefore, in 1 minutes 40 meters length of the water is dumped into the sea. The cube thus formed has dimensions 40 x 60x 3 = 7200 cu meter
7. If the volume of two cubes are in ratio 27:64, then the ratio of their total surface area is
1. 27:64 2. 3:4
3. 9:16 4. 3:18
Answer: 2. 3:4
Volume of cube 1 / Volume of cube2 = 27/64
=> length of cube1 / length of cube 2 = v27/64 = 3/4
The formula for the total surface area of a cube is 6 L
Therefore, the ratio of total surface area = 6l/6L = 6 (length of cube 1)/6(length of cube 2)
= (length of cube 1)/(length of cube 2)
= 3/4
8. The height of a wall is six times its width and the length of the wall is seven times its height. If volume of the wall be 16128 cu m. Its width is
1. 5m 2. 4m
3. 4.5 m 4. 6 m
Answer:2. 4m
Let the length of the wall be L and width W and height H, then
Given that ,
H = 6W
and L = 7H or
L = 7 x 6w = 42W
Again, the volume of a cube = H x L X W
16128 = 6W x 42W x W
=> W³ = (16128 )/(6 x 42)
W³ = 64
W = 4
Therefore width is 4 cm.
9. The volume of a cuboid is twice the volume of a cube. If the dimensions of the cuboid are 9 cm,8 cm and 6 cm. The total surface area of the cube is
1. 72 sq cm 2. 216 sq cm
3. 432 sq cm 4. 108 sq cm
Answer: 6 cm
Given that the volume of cuboid is twice that of a cube
volume of cuboid = 9 x 8 x 6 = 432
Thus the volume of cube is 432/2 = 216
And the length of the cube = ∛216 = 6 cm
And thus total surface area of cube = 6 x 6 x6 = 216 sq cm
10. The edges of a cuboid are in ratio 1:2:3 and its surface area is 88 cm² .
the volume of the cuboid is
1. 120 cm³ 2. 64 cm³
3. 48 cm³ 4. 24 cm³
Answer:3. 48 cm³
Let the edges of cube be L , 2L , 3L
And the surface area is 2(L x 2L) + 2(L + 3L) + 2(2L + 3L) = 88
4L² + 8L² + 10L² = 22L² = 88
=> L² = 4
=> L = 2 cm
And the volume is 2 x 2(2) x 3(2) = 2 x 4 x 6 = 48 cm³
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