Cube Volume to Side Length
The volume of a cube is given by the formula $V = s^3$, where $s$ is the side length.
We are given the volume $V = 1728\text{ cm}^3$. To find the side length $s$, we take the cube root:
$s = \sqrt[3]{1728\text{ cm}^3}$
Since $12 \times 12 \times 12 = 1728$, the side length of the cube is:
$s = 12\text{ cm}$
Sphere Dimensions from Cube
A sphere inscribed inside a cube has a diameter equal to the side length of the cube.
- Cube side length, $s = 12\text{ cm}$
- Sphere diameter, $d = s = 12\text{ cm}$
- Sphere radius, $r = \frac{d}{2} = \frac{12\text{ cm}}{2} = 6\text{ cm}$
Sphere Surface Area Calculation
The formula for the total surface area ($SA$) of a sphere is $SA = 4\pi r^2$.
Using the calculated radius $r = 6\text{ cm}$:
$SA = 4\pi (6\text{ cm})^2$
$SA = 4\pi (36\text{ cm}^2)$
$SA = 144\pi\text{ cm}^2$
Therefore, the total surface area of the sphere is $144\pi\text{ cm}^2$. This matches Option A.
Similar Questions
- A solid pyramid is to be painted. If paint covers 2 $m^2$/litre and total surface is 48 $m^2$, how much paint is needed, including 5% extra for retouch?
- A large wooden box, open at the top, needs to be lined with fabric on its inner floor and inner four side walls. The inner dimensions of the box are 2.5 m length, 1.8 m wide, and 1.2 m high. If the fabric comes in square pieces of side 50 cm, and 15% extra fabric is needed for cuts and overlaps, how many approximate fabric pieces should be purchased?
- A square of side 10 cm has a smaller square (of side 4 cm) removed from one side. What is the ratio of the remaining area to the original?
- A rectangular park is 60 m by 40 m. A semicircular pond of diameter 40 m is inside. What percent of the park's area does the pond occupy?
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- What is the area of the segment formed by a chord in a circle of radius 6 cm, if the angle subtended at the center is $120^{\circ}$?
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The angle of a sector is π/4 radians, and the radius of the circle is 8 cm. What is the area of the sector?
Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
Find the surface area of a sphere whose diameter is equal to 28 cm.
