UNIT 12. Circle.

Unit 12. Circle - vocabulary.

12.1 The definition of circle.

- A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior.

- In everyday use, the term “circle” may be used interchangeably to refer to either the:

1) boundary of the figure (= okrąg), or to

2) the whole figure including its interior (= koło);

- In strict technical usage, the circle in the former and the latter is called a disc.

- A circle can also be defined as the curve traced out by a point that moves so that its distance from a given point is constant.


12.2 Parts of a circle.




any connected part of the circle.



the point equidistant from th epoints on a circle.



a line segment whose endpoints lie on the circle.



the length of one circuit along the circle.



the longest chord, a line segment whose endpoints lie on the circle and which pases through the centre, or the length of such a segment, which is the largest distance between any two points on the circle.



a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.



an extended chord, a straight line cutting the circle at two points.



a region bounded by two radii and an arc lying between the radii.



a region bounded by a chord and an arc lying between the chord’s endpoints.



a region bounded by a diameter and an arc lying between the diameter’s endpoints. It is a special case of a segment.



a straight line that touches the circle at a single point.


12.3 Analytic results.

Length of circumference

The ratio of a circle’s circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

C = 2πr = πd 

Area enclosed

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius, which comes to π multiplied by the radius squared.

Area = πr2


12.4 Inscribed angle theorem.


An inscribed angle (examples are blue and green angles in the figure) is exactly half the corresponding central (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).



Unit 12.   Circle - reading.

 Reading #1      Brief history of the notion of "circle".

   The compass in this 13th-century manuscript is a symbol of God’s act of Creation. Notice also the circular shape of the halo.

The word “circle” derives from the Greek κίρκος (kirkos). The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus. Early science, particularly geometry, astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically “divine” or “perfect” that could be found in circles.

Some highlights in the history of the circle are:

  • * 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 (3.16049…) as an approximate value of π.
  • * 300 BC – Book 3 of Euclid’s Elements deals with the properties of circles.
  • * In Plato’s Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
  • * AD 1880 – Lindemann proves that π is transcendental effectively settling the millennia-old problem of squaring the circle.



Brief history of the notion of “circle”. Comprehension.


How is a part of a plant related to the development of the notion of “circle”?


What devices help incorporate the theoretical notion of “circle” into our everyday life?


Which sections of mathematics are closely related to the notion of “circle”?


Circle used to be attributed some supernatural power.  True or false?


The value of π has changed since 1700 BC. True or false?


Reading#2      Squaring the circle.


Squaring the circle; the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealised compass and straightedge. 

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann-Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number. That is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if pi were transcendental, but pi was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.

In mathematics, a transcendental number is a possibly complex number that is not algebraic – that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e.

The expression “squaring the circle” is sometimes used as a metaphor for trying to do the impossible.

The term quadrature of the circle is sometimes used synonymously or may refer to approximate or numerical methods for finding the area of a circle.



Squaring the circle. Comprehension. 


What does “squaring the circle” mean?


Why wasn’t it proved that squaring the circle is impossible using only a finite number of steps until the early eighties of the ninetieth century?


Explain the notion of “transcendental number”.


What does “squaring the circle” mean in everyday language? Make a sentence using the notion in this casual meaning.






Definitions, pictures and articles for reading adapted from: