Volume Comparison: Sphere vs. Cube

To determine how the volume of a sphere of diameter 1 unit compares to the volume of a cube of side 1 unit, we need to calculate the volume of each shape separately and then compare the results.

Sphere Volume Calculation

The formula for the volume of a sphere is given by:

$$V_s = \frac{4}{3} \pi r^3$$

Where \(V_s\) is the volume of the sphere and \(r\) is its radius.

• The question states the diameter of the sphere is 1 unit.
• The radius \(r\) is half of the diameter, so \(r = \frac{\text{Diameter}}{2} = \frac{1}{2}\) unit.

Substituting the value of the radius into the formula:

$$V_s = \frac{4}{3} \pi \left(\frac{1}{2}\right)^3$$

$$V_s = \frac{4}{3} \pi \left(\frac{1}{8}\right)$$

$$V_s = \frac{4\pi}{24}$$

$$V_s = \frac{\pi}{6} \text{ cubic units}$$

To get an approximate numerical value, we can use the common approximation for \(\pi \approx 3.14159\):

$$V_s \approx \frac{3.14159}{6}$$

$$V_s \approx 0.5236 \text{ cubic units}$$

Cube Volume Calculation

The formula for the volume of a cube is given by:

$$V_c = a^3$$

Where \(V_c\) is the volume of the cube and \(a\) is the length of its side.

• The question states the side of the cube is 1 unit.
• So, \(a = 1\) unit.

Substituting the value of the side into the formula:

$$V_c = (1)^3$$

$$V_c = 1 \text{ cubic unit}$$

Volume Comparison and Conclusion

Now, let's compare the calculated volumes of the sphere and the cube:

• Volume of the sphere ($V_s$) \(\approx 0.5236\) cubic units.
• Volume of the cube ($V_c$) \(= 1\) cubic unit.

Since \(0.5236 < 1\), it is clear that the volume of the sphere is numerically smaller than the volume of the cube.

Therefore, the volume of a sphere of diameter 1 unit is less than the volume of a cube of side 1 unit.