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
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
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 ______
