Generally to calculate the area of a bbox in a projected coordinate system since it's a (big) rectangle you can use the area formula :
area = (sw_longitude - ne_longitude) * (sw_latitude - ne_latitude)
Depending now on your spatial location (ie you're in a projected crs) the above formula will give you square mapunits (km^2, m^2 whatever).
In case you're in a sphere, like earth, you can use the sperical zone approach:
Where the area can be calculated with the following formula :
And since you want a sector of the zone, you'll need to multiply the above with where
α = lat2 - lat1
Thus the formula for the bbox area equals with: