Madhyamik Mathematics Chapter 4 – Cuboid focuses on important concepts and question-answer practice related to cuboid and cube. This chapter includes MCQs, True or False questions, fill-in-the-blanks, short-answer questions and long-answer numerical problems.
The practice set below has been reorganized in a clear and student-friendly format while keeping all the question categories from the source chapter together in one place.
Chapter 4: Cuboid
আয়তঘন | Madhyamik Mathematics Suggestion
MCQ
5 Questions
Objective
True/False + Fill in the Blanks
Written
Short + Long Questions
Basic Concepts of Cuboid
A cuboid is a three-dimensional solid whose six faces are rectangular. A cube is a special type of cuboid in which length, breadth and height are equal.
| Property |
Cuboid |
Cube |
| Faces |
6 |
6 |
| Edges |
12 |
12 |
| Vertices |
8 |
8 |
| Solid Diagonals |
4 |
4 |
Important Formulae
Cube
Volume = a³
Total Surface Area = 6a²
Face Diagonal = a√2
Solid Diagonal = a√3
Sum of All Edges = 12a
Cuboid
Volume = l × b × h
Total Surface Area = 2(lb + bh + hl)
Solid Diagonal = √(l² + b² + h²)
Sum of All Edges = 4(l + b + h)
Remember: The face diagonal of a cube is a√2, whereas its solid diagonal is a√3. This distinction is useful for both MCQ and numerical questions.
Multiple Choice Questions
MCQ – 1 Mark Each
1. If the diagonal of each face of a cube is 8√2 cm, what will be the length of the diagonal of the cube?
(a) 5√3 cm
(b) 6√3 cm
(c) 7√3 cm
(d) 8√3 cm
Answer: (d) 8√3 cm
2. A cuboid has x vertices, y edges and z faces. Find the value of x − y + z.
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (b) 2
3. If the diagonal of a cube is 5√3 cm, what will be the sum of the lengths of all its edges?
(a) 45 cm
(b) 55 cm
(c) 60 cm
(d) 65 cm
Answer: (c) 60 cm
4. If the total surface area of a cube is 150 cm², what will be its volume?
(a) 110 cm³
(b) 115 cm³
(c) 120 cm³
(d) 125 cm³
Answer: (d) 125 cm³
5. The longest rod that can be placed inside a cubical room is 5√3 metres long. What will be the area of the floor of the room?
(a) 16 m²
(b) 12√3 m²
(c) 4 m²
(d) 20 m²
Answer: The source lists the options above; use the cube diagonal relation to determine the required floor area.
True or False Questions
1. Two cubes placed side by side always form one large cube.
2. The total surface area of a cuboid is lb + bh + lh square units.
Answer: False. The total surface area is 2(lb + bh + lh).
3. The line segment where two faces of a cuboid intersect is called an edge.
Answer: False according to the source.
4. The points where the edges of a cuboid meet are called vertices.
Answer: False according to the source.
5. If each edge of a cuboid or cube increases by a%, its diagonal also increases by a%.
6. The diagonal of a cube is √6 times its edge.
7. If every edge of a cube is doubled, its volume becomes four times the original volume.
8. If rainfall of 5 cm occurs over 2 hectares of land, the volume of rainwater is 1000 cubic metres.
Source note: The True/False answers above are reproduced according to the supplied source page. Some statements on the source page appear to contain wording or answer-key inconsistencies, so they should be checked against the textbook before final publication.
Fill in the Blanks
1. The diagonal of one face of a cube is ______ times the length of one edge.
Answer: √2
2. The common part where two faces of a cuboid or cube meet is called the ______.
Answer: Edge
3. If each edge of a cube is 6 cm, its diagonal is ______ cm.
Answer: 6√3
4. The number of vertices of a cuboid is ______.
Answer: 8
5. If the edge of the first cube is twice the edge of the second cube, the ratio of their volumes is ______.
Answer: 8 : 1
6. Find the ratio between the face diagonal of a cube and its solid diagonal.
Answer: √2 : √3
7. If the diagonal of a cube is 4√3 units, its volume is ______ cubic units.
Answer: 64 cubic units
8. The number of diagonals of a cuboid is ______.
Answer: 4
9. The number of diagonals of a cuboid is ______.
Answer: 4
10. The diagonal of one face of a cube = ______ × length of one edge.
Answer: √2
11. If the length, breadth and height of a cuboid are equal, the solid is called a ______.
Answer: Cube
Short-Answer Questions
Question 1: Faces, Edges, Vertices and Diagonals
If a cuboid has x faces, y edges, z vertices and p diagonals, find the value of x − y + z + p.
For a cuboid:
x = 6, y = 12, z = 8, p = 4
Therefore,
x − y + z + p = 6 − 12 + 8 + 4 = 6
Answer: 6
Question 2: Volume of a Cube
The sum of all the edges of a cube is 96 cm. Find its volume.
Let each edge of the cube be
a cm.
12a = 96
Therefore,
a = 8 cm
Now,
Volume = a³ = 8³ = 512 cm³
Answer: 512 cm³
Question 3: Equal Volumes of Two Cuboids
The dimensions of two cuboids are respectively 4, 6, 4 units and 8, (2h − 1), 2 units. If their volumes are equal, find h.
For the first cuboid:
Volume = 4 × 6 × 4 = 96 cubic units
For the second cuboid:
Volume = 8 × (2h − 1) × 2
Equating the two volumes:
8 × (2h − 1) × 2 = 96
16(2h − 1) = 96
2h − 1 = 6
h = 7/2
Answer: h = 7/2
Question 4: Diagonal of a Cube
The total surface area of a cube is 600 cm². Find the length of its diagonal.
Let the edge of the cube be
a cm.
6a² = 600
Therefore,
a² = 100
So,
a = 10 cm
The solid diagonal is:
d = a√3 = 10√3 cm
Answer: 10√3 cm
Question 5: Equal Numerical Values of Volume and Surface Area
If the numerical values of the volume and total surface area of a cube are equal, find the length of its diagonal.
Let the edge be
a units.
Volume:
a³
Total surface area:
6a²
According to the condition:
a³ = 6a²
Since a is not zero:
a = 6
Therefore:
Diagonal = 6√3 units
Answer: 6√3 units
Question 6: Melting Three Cubes
Three solid cubes have edge lengths 3 cm, 4 cm and 5 cm respectively. They are melted together to form one new solid cube. Find the edge length of the new cube.
The combined volume is:
3³ + 4³ + 5³
= 27 + 64 + 125
= 216 cm³
Let the edge of the new cube be a cm.
a³ = 216
Therefore,
a = 6 cm
Answer: 6 cm
Long-Answer Questions
These are the 5-mark descriptive questions given in the source page. They are important because they require multiple calculation steps and proper application of cuboid and volume concepts.
Question 1: Filling a Rectangular Reservoir
If a pipe can fill 37,400 litres of water per hour, calculate how long it will take to fill a rectangular reservoir having the dimensions given in the problem up to a height of 17 decimetres.
According to the source problem, the reservoir dimensions used in the solution are:
Length = 18 m = 180 dm
Breadth = 11 m = 110 dm
Height = 17 dm
Therefore, the volume of the reservoir is:
Volume = 180 × 110 × 17 cubic dm
Since 1 cubic decimetre of water is equal to 1 litre:
Required water = 180 × 110 × 17 litres
Using the pump capacity given in the question, the source gives the required operating time as:
Time = 9 hours
Answer: The pump must be operated for 9 hours.
Question 2: Increase in Water Level
A rectangular tank is 2.1 metres long and 1.5 metres wide and is half-filled with water. If another 630 litres of water are added, by how much will the depth of the water increase?
Convert the additional water into cubic metres:
630 litres = 0.63 m³
The base area of the tank is:
Base Area = 2.1 × 1.5 = 3.15 m²
The increase in water level is found using:
Increase in depth = Added volume ÷ Base area
= 0.63 ÷ 3.15
= 0.2 m
Since 0.2 m = 2 dm:
Answer: The water level increases by 2 decimetres.
Question 3: Square Brass Plate
A square-based brass plate has side x cm and thickness 1 mm. Its mass is 4725 grams. If 1 cm³ of brass has a mass of 8.4 grams, find the value of x.
Thickness:
1 mm = 0.1 cm
Using mass and density, the volume of brass is:
Volume = Mass ÷ Density
= 4725 ÷ 8.4
= 562.5 cm³
For the square plate:
Volume = x² × 0.1
Therefore:
0.1x² = 562.5
x² = 5625
x = 75 cm
Answer: x = 75 cm
The source page presents this problem with an accompanying image and does not display the complete written solution in its text extraction. The calculation above follows directly from the numerical data visible in the source question.
Question 4: Increase in the Height of a Field
A rectangular field has length 20 metres and breadth 15 metres. Four cubical holes are made at the four corners for installing pillars, each having the given side length. The removed soil is spread over the remaining part of the field. Find the increase in the height of the field.
This problem is based on the principle that the volume of removed soil equals the volume of soil spread over the field.
Volume of removed soil = Volume of soil spread
First calculate the combined volume of the four cubical holes.
Volume of one cubical hole = a³
Volume of four holes = 4a³
The area over which the soil is spread is the remaining area of the field:
Remaining Area = 20 × 15 − 4a²
Therefore:
Increase in height =
4a³ ÷ (300 − 4a²)
The source page displays the value of the cubical-hole side through an image, and that numerical value is not present in the extracted text. Therefore the final numerical answer cannot be reliably reproduced from the text alone without inventing the missing value.
Chapter 4 Important Formula Chart
| Quantity |
Formula |
| Cube Volume |
a³ |
| Cube Total Surface Area |
6a² |
| Cube Face Diagonal |
a√2 |
| Cube Solid Diagonal |
a√3 |
| Sum of Cube Edges |
12a |
| Cuboid Volume |
lbh |
| Cuboid Total Surface Area |
2(lb + bh + hl) |
| Cuboid Diagonal |
√(l² + b² + h²) |
| Sum of Cuboid Edges |
4(l + b + h) |
Quick Revision Points
Cube: All three dimensions are equal.
Cuboid: Length, breadth and height may be different.
Cube faces: 6
Cube edges: 12
Cube vertices: 8
Cuboid diagonals: 4
Face diagonal of cube: a√2
Solid diagonal of cube: a√3
Cube volume: a³
Cuboid volume: lbh
How to Prepare Chapter 4
Focus on the Basic Formulae
Before attempting numerical questions, revise the formulas for volume, surface area, face diagonal, solid diagonal and sum of edges.
Practise Unit Conversion
The long questions often involve metres, decimetres, centimetres, millimetres and litres. Make sure that all measurements are converted into compatible units before applying a formula.
Practise Application-Based Problems
Questions involving reservoirs, water tanks, brass plates and fields are based on the same basic concepts of volume and area. Practising these applications can make the chapter much easier.
Exam Tip: For a 5-mark numerical problem, write the given data, formula, substitution, calculation and final answer separately. This makes the solution easier to follow and reduces calculation mistakes.
Chapter 4 at a Glance
Chapter 4 – Cuboid contains objective questions, short-answer problems and long numerical questions. The most important areas are cube and cuboid dimensions, diagonals, surface area, volume, ratios and real-life volume applications.
For complete preparation, practise all five MCQs, all eight True/False questions, all eleven fill-in-the-blanks, all six short-answer questions and all four long-answer questions given in this chapter.
