Was the British Empire bigger than the combined empires of the Iberian Union in terms of claimed land area?

This question is a little more involved than it might appear, and has a few layers we need to tear through to solve. The basis is obvious, if the Spanish Empire was huge, and the Portuguese Empire at one time was responsible for vast exploration and colonization as well, then at a time when these two empires combined, might we have seen the worlds largest empire?

• Did the Iberian Union(1580–1640), which combined the territories claimed by both Spain and Portugal, exceed in size the empire commonly accepted as the largest, the British Empire?

The first thing we can check is an entry listing the largest empires in Wikipedia. It gives some area numbers for each empire at its largest extent:

- British Empire        35.5 million km²    13.71 million mi²   23.84%  1920 

No entry for Iberian Union, but individually:

- Spanish Empire        13.7 million km²    5.29 million mi²    9.20%   1810
- Portuguese Empire     10.4 million km²    4.02 million mi²    6.98%   1815

At this point we can see that 24.1 million km² is way less than 35.5 million km², so question answered with one trivial visit to Wikipedia. But the page, even though it lists sources, does have a warning:

This section's factual accuracy is disputed.

So we need more information, or another way to tackle the question. The OP has a map from Wikipedia and a list of territories claimed at the time of the Iberian Union. It seems to show a huge amount of territory, and the distinction that we are looking into claimed territories is now part of the question definition.

Claimed can unfortunately be ambiguous as well, so perhaps a definition of Empire will help. Google gives us:

an extensive group of states or countries under a single supreme authority, formerly especially an emperor or empress.

The article providing our initial measurements says an empire is:

"any relatively large sovereign political entity whose components are not sovereign" and its size as the area over which the empire has some undisputed military and taxation prerogatives.

So from these definitions and the desire to qualify 'claimed' I will look at 'components' which have some form of definition as a political entity. Also to qualify as part of the Iberian Union these components must be defined(exist) within the time frame allotted: 1580–1640.

Most of the components in the list provided by the OP are relatively small, often well defined (islands) so not much debate concerning them. I will move on to the areas of contention, where most of the land mass lies, the Americas.

South America. The OP claims all of South America should fall into this empire, maps such as the one linked here show just a border region. If we look at treaties concerning the region, the Treaty of Tordesillas between Spain and Portugal split South America (and technically the world)

along a meridian 370 leagues west of the Cape Verde islands, off the west coast of Africa.

This essential split South America between the Spanish and the Portuguese. To qualify as part of the empire, we need to see a political structure, however, not just a 'claim', and this is seen here:

Spain established political divisions covering all of South America not covered by Portugal. By the basis of 'claim' and the presence of a political structure, I would include all of South America.

North America. This is the tough spot. Irregular lines and anachronistic maps found all over wikipedia make this difficult. The real question comes down to how much of North America can be (claimed and having some political structure) included in the Iberian Union.

No real debate concerning the areas of the Caribbean, Mexico or Central America. This means we need to figure out Spanish territory North of Mexico. I've cropped and expanded the OPs' map showing this region :

We can see extensive coverage by the Spanish Empire, but we need to look at dates. Spanish Louisiana looks huge, but:

Louisiana (Spanish: Luisiana) was the name of an administrative district of the Viceroyalty of New Spain from 1762 to 1802 that consisted of territory west of the Mississippi River basin, plus New Orleans. Spain acquired the territory from France,

This region was not part of the Spanish Empire during the time of the Iberian Union.

The upper region along the Canadian coast, the Nootka Territory, again dates show no political structure existed before the 18th century, so we can logically eliminate that region as a component of the Iberian Empire.

This leaves us with a 'fuzzy' border containing a region of the (now) western
United States, which may have fallen under undisputed claim by the Spanish during the Iberian Union, politically administered under the title of the Viceroyalty of New Spain.

The Numbers:

At this point, I believe we have narrowed the regions covered and the timing on most of the political structures to turn this into a geography question; add up the territories covered and compared to the British empire. I confess to getting much of the geographical info here just by googling, and have probably missed a few, but unless the figures are quite close I believe data covered will provide an adequate answer to the question.

European and African possessions:

Iberian Penninsula     ‪581,998 km²
Part of Italy          ‪301,337 km²
Sicily (25,711 km²‬),  Sardinia(‪24,089 km²‬),  Corsica(‪8,679 km²‬)
17 provinces of the Netherlands,        33,751 km²
small parts of France, Germany and Austria, (not itemized)
Portuguese Angola 1,246,700 km2, and Mozambique 801,590 km2
Cape Verde  4,033 km²,The Canaries 7,493 km², Madeira 800 km² ,the Azores 2,346 km²

subtotal=3,037,527 km²

New World & Pacific possessions:

South America  ‪‪     17,839,838‬ km²
Central America     507,967 km²
Mexico              1,964,247‬ km²
Cuba                108,780 km²
Hispaniola 76,179 km², Puerto Rico 9,103 km², Trinidad 5,130 km²
Philippines         343,448‬ km²

section total 20,854,692 km²

US region: (here to get a near approximation I will just combine the area of Florida and any state West of (and mostly west of) the western border of the Louisiana Purchase, as shown here. )

Florida 170,304 km², Louisiana 134,264 km² ,Texas 695,621 km²,
New Mexico 314,915 km²,Colorado 269,601 km²,Utah 219,887 km²,Idaho 216,630 km²
Arizona 295,254,Nevada 286,351,California 423,970

section subtotal 3,026,797 km²

So a grand total of 3,037,527 + 20,854,692 +3,026,797 = 26,919,016 km² ( ‪10,393,490 mi²)

Somewhat larger then the combined figures from the original wikipedia combined entry of 24.1 million km² , but not a challenger to the size of the British Empire at its greatest extent of 35.5 million km². A large enough difference that small discrepancies in any given value are not going to change the outcome.

So I conclude, from what figures and historical data I can find:

• The Iberian Union was not larger then the British Empire.