Question

How can you tell which atoms are in the same plane of cumulene? Why is allene preferentially twisted instead of planar?

Answer 1

Cumulenes are chemical compounds which are formed by carbon and hydrogen atoms with three or more consecutive double bonds. Allenes are formed with carbon atoms having propadiene as their parent compound.

Complete answer:
Cumulene structure is rigid in nature due to the formation of double bonds between the carbon atoms. All the carbon atoms are $sp$ hybridized in the structure which forms the double bond in cumulene. This $sp$ hybridization results in formation of two $\left( \pi \right)$ bonds, one with each neighbor carbon atom. These double bonds are perpendicular in nature with respect to carbon atoms and therefore leads to the formation of a linear geometry within the carbon chain.
Example of the simplest cumulene is butatriene with three consecutive double bonds.
Structure of cumulene is- butadiene

From the structure we clearly see that the entire structure of cumulene is present on the same plane.
In the allene the center carbon atom is associated with formation of two sigma bonds and two pi bonds. Hybridization of carbon atoms is $sp$ while the other two terminal carbon atoms are $s{p^2}$ hybridized. The bond angle is ${180^ \circ }$ which is formed by the three carbon atoms. This confers linear geometry but the terminal two carbon atoms have planar geometry but the bond angle is twisted ${90^ \circ }$ from each other.
Structure of allene is - $1,2 -$ propadiene

From the structure we clearly see that terminal hydrogen is twisted and out of the plane.
$\Rightarrow$ From the above discussion we conclude that cumulene is a planar molecule while allene is a twisted molecule.

Note:
Carbon suboxide is an example of cumulene. Cumulene are considered as cis-trans depending upon the number of double bonds. Cis-trans cumulene has odd number of double bonds in the compound. Allenes have a tendency to undergo the cyclo-addition reaction.