வட்டம்

கட்டற்ற கலைக்களஞ்சியமான விக்கிபீடியாவில் இருந்து.

யூக்கிளிட்டின் கேத்திர கணிதப்படி, ஒரு வட்டம் என்பது, ஒரு குறிப்பிட்ட புள்ளியொன்றிலிருந்து சம அளவான தூரத்தில், ஒரே தளத்திலுள்ள புள்ளிகளின் ஒழுக்கு(locus) ஆகும். குறிக்கப்பட்ட புள்ளி அவ் வட்டத்தின் "மையம்" எனவும், சம அளவான தூரம் அதன் "ஆரை" எனவும் அழைக்கப்படும். வட்டங்கள், அவ அமைந்துள்ள தளத்தை உட்புறம், வெளிப்புறம் என இரண்டாகப் பிரிக்கும் எளிமையான மூடிய வளைவுகளாகும். வட்டத்தின் சுற்றளவு "பரிதி" எனப்படும்.

ஆள்கூற்றுக் கேத்திர கணித முறையில், (x₀, y₀) என்பதால் குறிக்கப்படும் புள்ளியை மையமாகவும், r என்பதை ஆரையாகவும் கொண்ட வட்டமொன்றிலமைந்துள்ள (x, y) ஆல் குறிக்கப்படும் எல்லாப் புள்ளிகளும்

(x − x₀)² + (y − y₀)² = r² என்னும் சமன்பாட்டால் தரப்படும்.

வட்டத்தின் மையப் புள்ளி (0, 0) ஆக இருப்பின், இச் சமன்பாடு பின்வருமாறு அமையும்.

x² + y² = r².

(0, 0) வை மையமாகக் கொண்ட 1 அலகு ஆரையுடைய வட்டம் அலகு வட்டம் (unit circle) எனப்படும்.

The slope (or derivate) of a circle can be expressed with the following formula:

y´ = −x/y

எல்லா வட்டங்களும் ஒத்தவை; அதனால், ஒரு வட்டத்தின் சுற்றளவும், அதன் ஆரையும் விகிதசமனானவை, அதுபோலவே, வட்டத்தின் பரப்பளவும் அதன் ஆரையின் வர்க்கத்துக்கு விகிதசமனானது. இவ் விகிதசமனின் மாறிலிகள் முறையே 2πயும் πயுமாகும்.

இன்னொரு வகையில் சொல்வதானால்;

• வட்டமொன்றின் சுற்றளவு = 2 × π × (radius)
• வட்டத்தின் பரப்பளவு = π × (radius)²

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the two and dividing by 2, we get the area π r².

A line cutting a circle in two places is called a secant, and a line touching the circle in one place is called a tangent. The tangent lines are necessarily perpendicular to the radii, segments connecting the centre to a point on the circle, whose length matches the definition given above. The segment of a secant bound by the circle is called a chord, and the longest chords are those that pass through the centre, called diameters and divided into two radii. The area of a circle cut off by a chord is called a circle segment.

It is possible (Circle points segments proof) to find the maximum number of unique segments generated by running chords between a number of points on the perimeter of a circle.

If only (part of) a circle is known, then the circle's center can be constructed as follows: take two non-parallel chords, construct perpendicular lines on their midpoints, and find the intersection point of those lines.

A part of the circumference bound by two radii is called an arc, and the area (i.e., the slice of the disk) within the radii and the arc is a sector. The ratio between the length of an arc and the radius defines the angle between the two radii in radians.

Every triangle gives rise to several circles: its circumcircle containing all three vertices, its incircle lying inside the triangle and touching all three sides, the three excircles lying outside the triangle and touching one side and the extensions of the other two, and its nine point circle which contains various important points of the triangle. Thales' theorem states that if the three vertices of a triangle lie on a given circle with one side of the triangle being a diameter of the circle, then the angle opposite to that side is a right angle.

Given any three points which do not lie on a line, there exists precisely one circle whose boundary contains those points (namely the circumcircle of the triangle defined by the points). Given three particular points <(x₁,y₁), (x₂,y₂), (x₃,y₃)>, the equation of this circle is given in a simple way by this equation using the matrix (math) determinant:

A circle is a kind of conic section, with eccentricity zero. In affine geometry all circles and ellipses become (affinely) isomorphic, and in projective geometry the other conic sections join them. In topology all simple closed curves are homeomorphic to circles, and the word circle is often applied to them as a result. The 3-dimensional analog of the circle is the sphere.

Squaring the circle refers to the (impossible) task of constructing, for a given circle, a square of equal area with ruler and compass alone. Tarski's circle-squaring problem, by contrast, is the task of dividing a given circle into finitely many pieces and reassembling those pieces to obtain a square of equal area. Assuming the axiom of choice, this is indeed possible.

Three-dimensional shapes whose cross-sections in some planes are circles include spheres, spheroids, cylinders, and cones.

See also:

• Unit circle
• Descartes' theorem
• Isoperimetric theorem