The traveling circus was a popular event in 19th century America, and the “big top” was the main circus tent. The term “big tent” began to be used for the circus itself.
 
“Big top” was printed in the Brooklyn (NY) Union-Argus on April 27, 1880. “‘Big top,’ or circus tent” was printed in the Washington (DC) Post on May 11, 1883. “Big top—The great tent in which the performance is given. All tents are ‘tops’” was printed in a dictionary of circus lingo in the Cedar Rapids (IA) Evening Gazette on July 17, 1909.
 
     
Wikipedia: Circus
A circus is a company of performers who put on diverse entertainment shows that may include clowns, acrobats, trained animals, trapeze acts, musicians, dancers, hoopers, tightrope walkers, jugglers, magicians, unicyclists, as well as other object manipulation and stunt-oriented artists.
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From the late 18th to late 19th century, custom-made circus buildings (often wooden) were built with various types of seating, a centre ring, and sometimes a stage. The traditional large tents commonly known as “big tops” were introduced in the mid-19th century as touring circuses superseded static venues. These tents eventually became the most common venue. Contemporary circuses perform in a variety of venues including tents, theatres and casinos. Many circus performances are still held in a ring, usually 13 m (42 ft) in diameter.
 
(Oxford English Dictionary)
big top  n. (also with capital initials) orig. U.S. the main tent of a circus; (by metonymy) the circus; also in extended use.
1891   N.Y. Herald 26 Apr. 16/1 (headline)  Out on the Road with Barnum’s ‘Big Top’.
1946   Univ. of Chicago Mag. Dec. 9/2   The fact is that he was not really a good showman under the academic big-top.
           
27 April 1880, Brooklyn (NY) Union-Argus, pg. 3, col. 2:
AMUSEMENTS.
Barnum’s Greatest Show.
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All the space on the left side of the “big top” was thus reserved, and being fitted up with chairs instead of the ordinary board seats of the circus, proved a most desirable spot from which to witness the acts of men and beasts.
   
13 May 1883, Washington (DC) Post, pg. 8, col. 3: 
40 MILES WITH A CIRCUS.
WHAT A POST REPORTER SAW WHILE TRAVELING WITH BARNUM’S SHOW.
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By the same process, the “big top,” or circus tent, was staked off, each iron peg representing where a wooden stake or pole was to be driven.
 
Illinois Digital Newspaper Collections
17 July 1886, New York (NY) Clipper, “Variety, Minstrel and Circus,” pg. 279, col. 3:
A pleasant termination to the war of Forepaugh vs. Doris took place under the big top at Cambridge 10.
       
23 September 1889, Pittsburg (PA) Dispatch, “Managing a Circus,” pg. 3, col. 1:
In the big top or circus tent preparation was being made for to-day.
 
22 July 1890, Cleveland (OH) Plain Dealer, pg. 8, col. 5:
BARNUM & BAILEY’S
Properly Christened Greatest Show on Earth.
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The monster tent, called Big Top, is pitched on spacious grounds on Hough avenue near Willson.
 
27 February 1902, Franklin (PA) Evening News, “Lingo of the Circus,” pg. 4, col. 2:
The tents are called “tops” by circus men, and they are sub-divided into the “big top,” the “animal top,” the “kid top,” the “candy top,” and so on.
 
17 July 1909, Cedar Rapids (IA) Evening Gazette, “A ‘Dictionary’ of Circus Lingo,” pg. 15, col. 1:
Big top—The great tent in which the performance is given. All tents are “tops.”
     
11 February 1919, The Press and Banner (Abbeville, SC), “Technical Slang of the Circus Is Interpreted by Eddie Polo,” pg. 7, col. 1:
BIG-TOP—The big tent in which the principal circus performance takes place. Different from the smaller tents in which the side-show holds forth.
 
15 May 1921, New York (NY) Herald, “Gladys Walton Learns Circus Terms for Films,” sec. 3, pg. 6 , col. 6:
“Big top”—Tent in which main performance is held.
 
4 June 1923, Rutland (VT) Daily Herald, “Clown Interprets Circus Phrases,” pg. 16, col. 6:
Big top—Main circus tent.
 
OCLC WorldCat record
A day under the big top, a study in life and art,
Author: Irving K Pond
Publisher: [Chicago] Chicago literary club, 1924.
Series: On cover: Club papers ... Chicago literary club
Edition/Format:   Print book : English
   
OCLC WorldCat record
Circus. A girl’s own story of life under the “big top” ... Edited by Janet Mabie. [With plates.].
Author: Betty Boyd BELL; Janet MABIE
Publisher: Brewer, Warren & Putnam: New York, 1931.
Edition/Format:   Print book : English
 
OCLC WorldCat record
Big top recipes for little people : the Big Apple Circus official cookbook for kids and would-be clowns
Author: Big Apple Circus.
Publisher: Nashville, Tenn. : FRP, ©1999.
Edition/Format:   Print book : Juvenile audience : English
Summary:
“Ladies and Gentlemen, boys and girls, clowns, acrobats, and circus fans: Welcome to the Big Apple Circus’ big top tent for an inside look at ‘America’s One-Ring Wonder.’ Inside we will explore circus anecdotes, cooking for a circus, and fun recipes for children.”
 
OCLC WorldCat record
Fall of the big top : the vanishing American circus
Author: David Lewis Hammarstrom
Publisher: Jefferson, N.C. : McFarland & Co., ©2008.
Edition/Format:   eBook : Document : English
Summary:
“Explores circus history from 1793 to the present and addresses the cultural forces pushing big top shows toward “circus ballet.” Numerous photographs and in-depth interviews conducted with show owners, performers and directors reveal a contrast between circuses of yesterday and today, and honor the outstanding performers who created and sustained the appeal of the circus”—Provided by publisher.
 
Children’s Museum (Indianapolis, IN)
Why is a Circus Called a Big Top?
By: The Children’s Museum of Indianapolis
Posted On: January 31, 2017
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The Traveling Circus
As the circus grew in popularity, new circuses opened in countries like France and the United States. Typically, these circuses had permanent homes in large buildings. However, at that time, the United States didn’t have many cities that were big enough to support a permanent circus building. For this reason, circus owners created the traveling circus.
 
In 1825, Joshuah Purdy Brown, a circus entrepreneur, became the first to use a large canvas tent instead of a permanent building. As traveling circuses expanded, tent sizes grew to accommodate more rings for different acts. The large tent that housed numerous rings and stages became known as the “The Big Top”. This name stuck and became interchangeable with circus!