Short answers, pulled from the story.
A circle is a shape consisting of all points in a plane that are at a given distance from a given point called the centre. The distance between any point on the circle and the centre is known as the radius.
In 1882 CE Ferdinand von Lindemann proved that pi is transcendental which demonstrated that squaring the circle cannot be performed with straightedge and compass. This proof relied on the Lindemann-Weierstrass theorem to show that no sequence of geometric operations can produce the required square area.
Archimedes proved in his Measurement of a Circle that the area enclosed equals that of a triangle with a base length equal to the circle's circumference and height equal to the radius. He applied this fact to approximate pi using inscribed and circumscribed polygons.
Religious traditions use the circle to represent heavenly bodies divine spirits sacred concepts including unity infinity wholeness the universe divinity balance stability and perfection. A 13th-century manuscript shows the compass as a symbol of God's act of Creation while halos surrounding figures depict circular shapes.