### How do you calculate cross sectional area of wire for young's modulus?

You could use a micrometer to measure its diameter. Preferably
measure at several locations and use the average.
Then cross-sectional area = (1/4)*pi*d2
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Here is a cheaper alternative:
A=V/L
where,
A= the cross-sectional area of the wire
V= the volume of the wire, measured using water
displacement.
L= the length of the wire, measured using a ruler
also if the wire gauge is known simply use the following formula
to obtain diameter:
D=.005*92^((36-gauge#)/39)
As far as young's modulus calculation goes, here is the general
procedure:
Catalog the force applied to both ends of the wire with time.
use a strain gauge to measure the axial strain with time. calculate
the engineering stress using S=F/A.
plot the stress verses axial strain and then isolate the elastic
portion of the plot. This is the portion of the plot where the
slope is constant. The slope can be taken as Young's Modulus.