Volume Comparison: Sphere vs. Cube
To determine the relationship between the volume of a sphere and a cube, we calculate their volumes based on the given dimensions.
Calculating Sphere Volume
The sphere has a diameter of 1 unit.
- Radius ($r$): Diameter / 2 = 1 / 2 = 0.5 units.
- Volume of a sphere formula: $V_{sphere} = \frac{4}{3}\pi r^3$
- Substituting the radius:
- Calculation:
- Approximate value: $V_{sphere} \approx \frac{3.14159}{6} \approx 0.5236$ cubic units.
Calculating Cube Volume
The cube has a side length of 1 unit.
- Side length ($s$): 1 unit.
- Volume of a cube formula: $V_{cube} = s^3$
- Substituting the side length:
- Volume: $V_{cube} = 1$ cubic unit.
Comparing Volumes
Compare the calculated volumes:
- Sphere Volume ($V_{sphere}$) $\approx 0.5236$ cubic units.
- Cube Volume ($V_{cube}$) = 1 cubic unit.
Since $0.5236 < 1$, the volume of the sphere is less than the volume of the cube.
Conclusion
The correct comparative term is 'less'. The volume of the sphere (diameter 1 unit) is less than the volume of the cube (side 1 unit).
Important Questions from Mensuration and Geometry
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
Right triangle PQR is to be constructed in the xy - plane so that the right angle is at P and line PR is parallel to the x-axis. The x and y coordinates of P, Q, and R are to be integers that satisfy the inequalities: $-4\le x \le 5$ and $6 \le y \le 16$. How many different triangles could be constructed with these properties?
