Question

Find the surface area of the biggest sphere that can fit inside a cube of side \[4a\] cm.

Answer 1

Hint: The surface area of any given object is the area or region occupied by the surface of the object. Whereas volume is the amount of space available in an object. Hence, to find the surface area we need to find the value of radius, then we need to apply the formula \[4\pi {r^2}\] to get the surface area.
Formula used:
Surface area of the sphere is \[ = 4\pi {r^2}\]

Complete step-by-step solution:
Let us write the given data:
\[side = 4a\]\[cm\].
And the Surface area of the sphere is \[ = 4\pi {r^2}\]

The diameter of the biggest sphere will be equal to the side of a square, hence we get the radius as:
\[radius = \dfrac{{4a}}{2}\]
\[ \Rightarrow radius = 2a\]
We know that, the surface area of the sphere is given by \[4\pi {r^2}\].
Hence, substituting the value of radius we get the surface of the sphere as:
\[ \Rightarrow 4\pi {\left( {2a} \right)^2}\]
\[ \Rightarrow 4\pi \left( {4{a^2}} \right)\]
\[ \Rightarrow 16\pi {a^2}\]
Therefore, the surface area of the biggest sphere that can fit inside a cube of side 4a cm is \[16\pi {a^2}\].

Additional information: Surface area and volume are calculated for any three-dimensional geometrical shape.
Total surface area refers to the area including the base and the curved part. Curved surface area refers to the area of only the curved part of the shape excluding its base. It is also referred to as the lateral surface area for shapes such as a cylinder. The amount of space, measured in cubic units, that an object or substance occupies is called volume. Two-dimensional space doesn't have volume but has area only.

Note: We must note that each shape has its surface area as well as volume. Volume of Circle cannot be found, though Volume of the sphere can be, because a sphere is a three-dimensional shape. It is the total area covered by the surface of the object and if the shape has a curved surface and base, then total area will be the sum of the two areas.