Question

Find the volume of the biggest cone that can fit inside a cube of side $ 5\,cm $

Answer 1

Hint: In this problem we first find radius and height of the cone. As, biggest cone is placed inside the cube therefore the diameter of the base circle of the cone and height of the cone will be equal to the side of the cube and using these values in formula we can easily get the volume of the cone. Volume of a cone = $ \dfrac{1}{3}\pi {r^2}h $ , where ‘r’ and ‘h’ are radius and height of the cone.

Complete step-by-step answer:
When the biggest cone is placed in a cube. Then the base circle of the cone will cover the maximum area of the base face of the cube. Hence the circle of cones will touch all four sides of the square base of the cube.

Which implies diameter of circle of base of cone will be equal to side of square of base of cube.
Hence, from above we see that diameter of cone = $ 5\,cm $
Therefore, radius of cons is half of diameter = $ 2.5\,cm $
Also, placing a biggest cone in a cube of side $ 5\,cm $ , top vertex of the cone will touch the upper surface of the cube. Then the height of the cone will be equal to the height of the cube.
Therefore, from above we see that the height of the biggest cone which is placed inside a cube = $ 5\,cm $ .
Now, we can find the volume of the biggest cone that is placed inside the cube as we got both radius and height.
$\Rightarrow Volume\,\,of\,\,the\,\,biggest\,\,cone = \dfrac{1}{3}\pi {r^2}h $
Substituting $ r = 2.5\,\,and\,\,h = 5 $ in above written formula
Volume of cone $ = \dfrac{1}{3} \times \dfrac{{22}}{7} \times 2.5 \times 2.5 \times 5 $
Volume of biggest cone = $ 32.73 $
Hence, from above we see that volume of biggest cone that is placed inside a cube of side $ 5\,cm $ is $ 32.73\,c{m^3} $

Note: When the biggest cone is placed inside a cube. It means that the base of the cone covers the maximum area of the base of the cube. So, we can’t consider that cone whose base circle is not touching all four sides of the base square of the cube. Also, the biggest cone height will be such that its top vertex will touch the upper square surface of the cube. Hence, the height of the biggest cone must be the same as the height of the cube. So, if one considers any other type of cone it will not be considered as biggest and leads to wrong answers.