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, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
