Find the total surface area of a cube whose volume is 343 m3.
294 m2
Finding the Total Surface Area of a Cube Given Its Volume
This problem requires us to find the total surface area of a cube when we are given its volume. To solve this, we first need to find the side length of the cube using the volume formula. Once we have the side length, we can use the formula for the total surface area of a cube.
Understanding Cube Formulas
A cube is a three-dimensional shape with six equal square faces. All edges of a cube have the same length.
- The volume ($V$) of a cube with side length ($s$) is given by the formula: $$V = s^3$$
- The total surface area ($A$) of a cube with side length ($s$) is the sum of the areas of its six square faces. Each face has an area of $s^2$. So, the total surface area is: $$A = 6s^2$$
Step-by-Step Calculation
We are given that the volume of the cube is $343 \text{ m}^3$. Let the side length of the cube be $s$ meters.
Step 1: Find the side length of the cube
Using the volume formula, we have:
$$V = s^3$$ $$343 \text{ m}^3 = s^3$$To find $s$, we need to take the cube root of 343:
$$s = \sqrt[3]{343} \text{ m}$$We need to find a number that, when multiplied by itself three times, equals 343. Let's check some small integer values:
- $5 \times 5 \times 5 = 25 \times 5 = 125$ (Too small)
- $6 \times 6 \times 6 = 36 \times 6 = 216$ (Too small)
- $7 \times 7 \times 7 = 49 \times 7 = 343$ (Correct!)
So, the side length of the cube is $s = 7$ meters.
Step 2: Calculate the total surface area of the cube
Now that we have the side length $s = 7$ m, we can use the total surface area formula:
$$A = 6s^2$$Substitute the value of $s$ into the formula:
$$A = 6 \times (7 \text{ m})^2$$ $$A = 6 \times (49 \text{ m}^2)$$ $$A = 294 \text{ m}^2$$The total surface area of the cube is $294 \text{ m}^2$. This matches one of the given options.
Comparing with Options
Let's compare our calculated total surface area with the given options:
| Option | Surface Area Value |
|---|---|
| 1 | 186 m<sup>2</sup> |
| 2 | 294 m<sup>2</sup> |
| 3 | 196 m<sup>2</sup> |
| 4 | 210 m<sup>2</sup> |
Our calculated value, $294 \text{ m}^2$, matches Option 2.
Revision Table: Cube Formulas
| Property | Formula (side = s) |
|---|---|
| Volume | $V = s^3$ |
| Total Surface Area | $A = 6s^2$ |
| Lateral Surface Area (Area of 4 sides) | $A_{\text{lateral}} = 4s^2$ |
| Diagonal of one face | $d_{\text{face}} = s\sqrt{2}$ |
| Space diagonal of the cube | $d_{\text{space}} = s\sqrt{3}$ |
Additional Information: Properties of a Cube
Beyond volume and surface area, a cube has several interesting properties:
- A cube is a special type of rectangular prism where all edges are equal.
- It is one of the five Platonic solids.
- It has 6 faces (squares), 12 edges (all equal length), and 8 vertices.
- Opposite faces are parallel.
- Adjacent faces are perpendicular.
Understanding these basic geometry concepts and formulas is crucial for solving problems involving cubes and other 3D shapes.
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