What is the volume of a cube if the perimeter of one face of the cube is 40 cm?
1000 cm3
Understanding the Problem: Cube Volume Calculation
The question asks us to find the volume of a cube given the perimeter of one of its faces. A key piece of information here is that a face of a cube is always a square. The volume of a cube depends on the length of its side.
Step-by-Step Solution to Find Cube Volume
Here's how we can solve this problem:
- Identify the shape of the face: A cube's face is a square.
- Recall the formula for the perimeter of a square: The perimeter of a square is given by $P = 4 \times \text{side}$, where 'side' is the length of one side of the square.
- Use the given perimeter to find the side length of the face: We are given that the perimeter of one face is 40 cm. Using the formula:
$\text{Perimeter} = 4 \times \text{side}$
$40 \text{ cm} = 4 \times \text{side}$
To find the side, we divide the perimeter by 4:
$\text{side} = \frac{40 \text{ cm}}{4}$
$\text{side} = 10 \text{ cm}$
So, the length of one side of the square face is 10 cm.
- Understand the relationship between face side and cube side: The side length of the square face is the same as the side length (or edge length) of the cube. Therefore, the side length of the cube is 10 cm.
- Recall the formula for the volume of a cube: The volume of a cube is given by $V = \text{side}^3$, where 'side' is the length of the cube's edge.
- Calculate the volume of the cube: Using the side length we found (10 cm):
$V = (10 \text{ cm})^3$
$V = 10 \text{ cm} \times 10 \text{ cm} \times 10 \text{ cm}$
$V = 1000 \text{ cm}^3$
Final Answer Derivation
Based on our calculation, the volume of the cube with a face perimeter of 40 cm is 1000 cm$^3$. This matches one of the provided options.
| Step | Description | Calculation |
|---|---|---|
| 1 | Perimeter of face (square) | 40 cm |
| 2 | Formula for square perimeter | $P = 4 \times \text{side}$ |
| 3 | Calculate side of square face | $\text{side} = \frac{P}{4} = \frac{40}{4} = 10 \text{ cm}$ |
| 4 | Side of cube | 10 cm |
| 5 | Formula for cube volume | $V = \text{side}^3$ |
| 6 | Calculate cube volume | $V = (10 \text{ cm})^3 = 1000 \text{ cm}^3$ |
Revision Table: Cube Formulas
| Property | Formula (s = side length) |
|---|---|
| Perimeter of a face (square) | $4s$ |
| Area of a face (square) | $s^2$ |
| Total Surface Area | $6s^2$ |
| Volume | $s^3$ |
| Diagonal of a face | $s\sqrt{2}$ |
| Space diagonal of cube | $s\sqrt{3}$ |
Additional Information: Properties of a Cube
A cube is a special type of rectangular prism where all edges are equal in length. It is one of the five Platonic solids. Understanding the properties of a cube is essential for solving geometry problems related to volume, surface area, and dimensions.
- A cube has 6 faces, and each face is a square.
- A cube has 12 edges of equal length.
- A cube has 8 vertices.
- All angles in a cube are right angles ($90^\circ$).
- The relationship between the perimeter of a face and the volume involves two main steps: first finding the side length from the perimeter, and then cubing the side length to find the volume.
Problems like this test your understanding of basic geometric shapes and how different properties (like perimeter and volume) are related through side lengths.
Similar Questions
If the total surface area of a cube is 24 sq.units, then what is the volume of the cube?
The volume of a cone is 73920 cm3. If the height of the cone is 160 cm, then find the diameter of its base.
Ranu carries water to school in a cylindrical flask with diameter 12 cm and height 21 cm. Determine the amount of water that she can carry in the flask. (Use π = \(\frac{22}{7}\))
The volume of a cone with height equal to radius, and slant height 5 cm is :
What is the whole surface area of a cone of base radius 6 cm and height 8 cm?
A spherical ball of lead, 3 cm in diameter, is melted and recast into three spherical balls. The diameters of two of these balls are \(\frac{3}{2}\) cm and 2 cm, respectively. Find the diameter of the third ball.
A conical tent of height 10 m and base diameter 48 m was erected by a company in a park. Find the curved surface area of the tent (In m2).
If the surface area of a cube is 5046 cm2, then the volume of the cube is:
The volume of a cuboid is twice that of a cube. If the dimensions of the cuboid are (8 m × 8 m ×16 m), the total surface area of the cube is:
If the slant height of a cone is 60 cm and the radius of its base is 21 cm, then find its curved surface area. (use π = \({22 \ {} \over 7}\))
Important Questions from Solid Figures
The area of the floor of a cubical room is 192 m 2. The length of the longest rod that can be kept in that room is :
A solid metallic rectangular block of dimensions 112 cm × 44 cm × 25 cm is melted and recast into a cylinder of radius 35 cm. The curved surface area (in cm 2) of the cylinder is: (Take π = 22/7)
If the volume of a cube is 175616 cm 3, what is its side?
The volume of a right circular cone is 1232 cm 3. If the height of the cone is 24 cm, then what will be the radius of its base?
A right triangle contains the right angle between the sides 5 cm and 7 cm. A cone is generated by revolving about the side 5 cm. The volume of this cone is:
