Question

Calculate the volume of a regular octahedron whose edges are all 10cm.

Answer 1

Hint: We know that there are many shapes and polygons in geometry. Among them, a regular octahedron is one of the polyhedral in platonic solids. Platonic solids play a vital role in our real life applications. A regular octahedron which can be simply called the octahedron consists of twelve edges, six vertices and eight edges. In plural, octahedron can be called octahedra. Here, each and every face of the octahedron is an equilateral triangle.
Now, we need to calculate the volume of a regular octahedron whose edges are given. We can easily find the required answer by simply substituting the length of the edges in the known formula. Here, the length of the edges is given as $10cm$

Formula used:
The formula to calculate the volume of a regular octahedron for the given edges is as follows.
Volume of a regular octahedron, $V = \dfrac{{\sqrt 2 }}{3}{a^3}$
Where, $a$ is the given length of the edge.

Complete step-by-step solution:
It is given that, the edges of a regular octahedron are$10cm$
That is,
$a = 10cm$
Now, substitute the value of edge in the formula.
Volume of a regular octahedron,
$V = \dfrac{{\sqrt 2 }}{3}{a^3}$
\[\Rightarrow V = \dfrac{{\sqrt 2 }}{3}{\left( {10} \right)^3}\]
\[\Rightarrow V = \dfrac{{1.414}}{3}\left( {1000} \right)\] (Rewriting $\sqrt 2 $ as $1.414$ )
 \[\Rightarrow V = \dfrac{{1414}}{3}\]
\[\Rightarrow V = 471.3c{m^3}\]
Therefore, the volume of a regular octahedron whose edges are all $10cm$ is $471.3c{m^3}$ . That is the required answer.

Note: A regular octahedron is one of the polyhedral in platonic solids. Platonic solids play a vital role in our real life applications. A regular octahedron consists of eight equal equilateral triangles. It contains six vertices and at each vertex, four edges meet which is an amazing fact.