Magic Circle

A Magic Circle (魔方陣) is a design or arrangement constructed to facilitate the use of Magic.

Principles
There are various elements which can be used within a magic circle and various forms which they can take - the simplest and first form being that of a lone circle. As the amount of information given to a magic circle has a direct relation with its power, magical symbols like the pentagram, the hexagram, the, or are drawn within the circle for additional effects. Other things and words are written along the circumference of the circle to draw power from what they represent via Idol Theory, like the name of the angel whom the power is borrowed from, the type of element, the amount and quality of Telesma needed and how it's going to be used, etc.

However, as a result, the more complicated a circle is, the more difficult it is to control it. If there's a difference between what the caster intended and what is interpreted incorrectly on the magic circle, the spell will go out of control.

Magic circles can be created through various means, from drawing designs and arranging certain objects, to using the positions of the stars themselves. Although magic circles are often constructed on two-dimensional surfaces, they can also be constructed in three-dimensional space.

Index Arc
After being injured by Stiyl Magnus, who was unaware that her protective Walking Church had been broken, Index drew a magic circle on a table using her blood and guided Tsukuyomi Komoe in casting healing magic on her wounds, saving her life. The magic circle had a pentagram in its center.

A unicursal hexagram was one of the elements of the spell which was going to be used to erase Index's memory. Magic circles appeared in Index's eyes while she was in John's Pen Mode after the Collar was touched by Imagine Breaker.

Angel Fall Arc
Tsuchimikado Motoharu mentioned Tactical Circles, a special type of Magic Circle, that are used for Grand Magic.

Kazakiri Hyouka Arc
During her attack on Academy City, Sherry Cromwell drew a number of magic circles with her oil pastel chalk, connected to Golem Ellis. In her final clash with Touma, she intended to use the circles drawn through the underground area to collapse it on her enemy but was forced to abort her attack when Touma identified and destroyed the magic circle which would have kept her safe.

Orsola Aquinas Rescue Arc Arc
During the battle with the Agnese Forces at the Church of Orsola, the Amakusas placed Stiyl Magnus's rune cards in specific positions across the church's grounds in order to form a large magic circle and enhance the power of Innocentius.

Toaru Majutsu to Kagaku no Ensemble
During Pantagruel's attempt to summon Aleister Crowley using the Silver Star Miracle, a magic circle (specifically a textless version of the, consisting of a pentagram, two heptagons, a heptagram and a pair of concentric circles) formed in the sky over the city.

Daihasei Festival Arc
During the events of the Daihaseisai and their pursuit of Oriana Thomson, Stiyl Magnus and Tsuchimikado Motoharu explained the basic principles behind magic circles and grimoires to Kamijou Touma.

British Royal Family Arc
During the British Halloween, the Anglicans made use of the special railway car on the underground line below Buckingham Palace, with a magic circle specifically designed as a safety device to stabilize the Curtana in the event of the sword's power going out of control, in conjunction with a flash spell from the Coven Compass to cause Carissa's Curtana Original to overload.

While heading for London on horseback during the coup, Elizard thought about the magic circles in the roads which could have boosted her horse's power and shortened her journey, but being under the jurisdiction of the British government would have alerted Carissa and the Knights to her location if she used them. Later, during the Battle of Buckingham Palace, the captive Bayloupe used a marker pen to draw a communication magic circle to contact the other members of New Light.

World War III Arc
During the later stages of Project Bethlehem, Sasha Kreutzev carved a magic circle to act as a relay point for spells being released from the surface against the Star of Bethlehem, aided by Lessar and the Russian Orthodox magicians onboard the fortress.

Kamisato Rescue Arc
Toyama Luca made use of a unicursal hexagram, drawn through the roads of Academy City with spilled gunpowder while her group were fleeing from the Kamisato Faction and ignited with a musket spark, in the spell used by her and Karasuma Fran to rescue Kamisato Kakeru from the 'new world'.

Necromancer Arc
Numerous magic circles, with elements of both Western Kabbalah and Eastern Taoism, were seen through uses of Rosenthal-style magic during the events surrounding DA, the Hishigata siblings, the Coffins and the Rosenthal Family.

Trivia

 * One particular magic circle is featured prominently in the anime and manga adaptations of Toaru Majutsu no Index, as well as their promotional materials and related media. It consists of a circle containing a pentagram, with a smaller circle circumscribed by the center of the pentagram. Several sets of letters are arranged around the perimeter of the circle, with crosses separating them. In some versions, two larger concentric circles surrounding the base circle and letters. The sequence of letters is as follows (clockwise); "ISCATC+HFIMMMP+MNIP+MNIA+ICY+ISWS".