Difference between revisions of "Circle"

m (Proof Using Calculus)
(Improved organization. Also, more theorems!)
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{{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
 
{{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  
== Traditional Definition ==
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==Definition==
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=== Traditional Definition ===
 
A circle is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance from a fixed point.  The fixed point is called the [[center]] and the distance from the center to a point on the circle is called the [[radius]].  
 
A circle is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance from a fixed point.  The fixed point is called the [[center]] and the distance from the center to a point on the circle is called the [[radius]].  
 
[[Image:circle1.PNG|thumb|right|The radius and center of a circle.]]
 
[[Image:circle1.PNG|thumb|right|The radius and center of a circle.]]
  
== Coordinate Definition ==
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=== Coordinate Definition ===
 
Using the traditional definition of a circle, we can find the general form of the equation of a circle on the [[coordinate plane]] given its radius, <math>r</math>, and center <math>(h,k)</math>.  We know that each point, <math>(x,y)</math>, on the circle which we want to identify is a distance <math>r</math> from <math>(h,k)</math>.  Using the [[distance formula]], this gives <math>\sqrt{(x - h)^2 + (y - k)^2} = r</math> which is more commonly written as
 
Using the traditional definition of a circle, we can find the general form of the equation of a circle on the [[coordinate plane]] given its radius, <math>r</math>, and center <math>(h,k)</math>.  We know that each point, <math>(x,y)</math>, on the circle which we want to identify is a distance <math>r</math> from <math>(h,k)</math>.  Using the [[distance formula]], this gives <math>\sqrt{(x - h)^2 + (y - k)^2} = r</math> which is more commonly written as
 
<cmath>(x - h)^2 + (y - k)^2 = r^2.</cmath>
 
<cmath>(x - h)^2 + (y - k)^2 = r^2.</cmath>
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<center>[[Image:Circlecoordinate1.PNG]]</center>
 
<center>[[Image:Circlecoordinate1.PNG]]</center>
  
== Area of a Circle ==
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==Circumference and Area==
The area of a circle is <math>\pi r^2</math> where <math>\pi</math> is the mathematical constant [[pi]] and <math>r</math> is the radius.
 
  
=== Archimedes' Proof ===
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Given a circle of radius <math>r</math>, the [[circumference]] (distance around a circle) is <math>2 \pi r</math> and the area is <math>\pi r^2</math>.  Both formulas involve the mathematical constant [[pi]] (<math>\pi</math>).
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=== Archimedes' Proof of Area===
 
We shall explore two of the Greek [[mathematician]] [[Archimedes]] demonstrations of the area of a circle.  The first is much more intuitive.
 
We shall explore two of the Greek [[mathematician]] [[Archimedes]] demonstrations of the area of a circle.  The first is much more intuitive.
  
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{{stub}}
 
{{stub}}
===Proof Using Calculus===
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===Area Proof Using Calculus===
 
Let the circle in question be <math>x^2 + y^2 = r^2</math>, where r is the circle's radius. By symmetry, the circle's area is four times the area in the first quadrant. The area in the first quadrant can be computed using a definite integral from 0 to r of the function <math>f(x) = \sqrt{r^2 - x^2}</math>. Using the substitution <math>x = r \sin u, dx = r \cos u</math> gives the indefinite integral as <math>\frac{r^2}{2} (u - \frac{\sin 2u}{2}) + C</math>, so the definite integral equals <math>\frac{r^2}{2} * \frac{\pi}{2}</math>. Multiplying by four gives the area of the circle as <math>\pi r^2</math>.
 
Let the circle in question be <math>x^2 + y^2 = r^2</math>, where r is the circle's radius. By symmetry, the circle's area is four times the area in the first quadrant. The area in the first quadrant can be computed using a definite integral from 0 to r of the function <math>f(x) = \sqrt{r^2 - x^2}</math>. Using the substitution <math>x = r \sin u, dx = r \cos u</math> gives the indefinite integral as <math>\frac{r^2}{2} (u - \frac{\sin 2u}{2}) + C</math>, so the definite integral equals <math>\frac{r^2}{2} * \frac{\pi}{2}</math>. Multiplying by four gives the area of the circle as <math>\pi r^2</math>.
  
==Related Formulae==
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==Lines in Circles==
* The [[area]] of a circle with radius <math>r</math> is <math>\pi r^2</math>.
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<asy>draw(unitcircle);draw((-0.8,1)--(1,1),Arrow);draw((1,1)--(-0.8,1),Arrow);draw((0,1)--(1,0));</asy>
* The [[circumference]] of a circle with radius <math>r</math> is <math>2\pi r</math>.
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A line that touches a circle at only one point is called the [[Tangent (Geometry)|tangent]] of that circle. Note that any point on a circle can have only one tangent.
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A line segment that has endpoints on the circle is called the [[chord]] of the circle. If the chord is extended to a line, that line is called a secant of the circle.  The longest chord of the circle is the diameter; it passes through the center of the circle.
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When two secants intersect on the circle, they form an [[inscribed angle]].
  
==Other Properties and Definitions==
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===Properties===
{{asy image|<asy>draw(unitcircle);draw((-0.8,1)--(1,1),Arrow);draw((1,1)--(-0.8,1),Arrow);draw((0,1)--(1,0));</asy>|right|A circle with a tangent and a chord marked.}}
 
*A line that touches a circle at only one point is called the [[Tangent (Geometry)|tangent]] of that circle. Note that any point on a circle can have only one tangent.
 
*A line segment that has endpoints on the circle is called the chord of the circle. If the chord is extended to a line, that line is called a secant of the circle.
 
 
*The measure of an [[inscribed angle]] is always half the measure of the [[central angle]] with the same endpoints.
 
*The measure of an [[inscribed angle]] is always half the measure of the [[central angle]] with the same endpoints.
*Chords, secants, and tangents have the following properties:
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**Since the diameter divides the circle into two equal parts, any angle formed by the two endpoints of a diameter and a third distinct point on the circle as the vertex is a right angle.
**The perpendicular bisector of a chord is always a diameter of the circle.
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**Also, a right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
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*Similarly, if a tangent line and a secant line intersects at the point of tangency, the measure of the angle formed is always half the measure of the [[central angle]] with the same endpoints.
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**From that property, the angle formed by the diameter and a tangent line with the point of tangency on the diameter is a right angle.
 
**The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
 
**The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
**The [[Power of a point]] theorem.
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*The perpendicular bisector of a chord is always a diameter of the circle.
Other interesting properties are:
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*When two chords <math>AB</math> and <math>CD</math> intersect at point <math>P</math> inside the circle, <math>\angle APC = \frac{m\widehat{AC} + m\widehat{BD}}{2}</math>.
*A right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
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*When two chords <math>AB</math> and <math>CD</math> intersect at point <math>P</math> outside the circle, <math>\angle APC = \frac{m\widehat{AC} - m\widehat{BD}}{2}</math>.
*Any angle formed by the two endpoints of a diameter of the circle and a third distinct point on the circle as the vertex is a right angle.
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*Lengths of chords can be calculated by using the [[Power of a point]] theorem.
  
 
==Problems==
 
==Problems==

Revision as of 15:27, 12 June 2019

A circle is a geometric figure commonly used in Euclidean geometry.

[asy]unitsize(2cm);draw(unitcircle,blue);[/asy]

Enlarge.png
A basic circle.


Definition

Traditional Definition

A circle is defined as the set (or locus) of points in a plane with an equal distance from a fixed point. The fixed point is called the center and the distance from the center to a point on the circle is called the radius.

The radius and center of a circle.

Coordinate Definition

Using the traditional definition of a circle, we can find the general form of the equation of a circle on the coordinate plane given its radius, $r$, and center $(h,k)$. We know that each point, $(x,y)$, on the circle which we want to identify is a distance $r$ from $(h,k)$. Using the distance formula, this gives $\sqrt{(x - h)^2 + (y - k)^2} = r$ which is more commonly written as \[(x - h)^2 + (y - k)^2 = r^2.\]

Example: The equation $(x - 3)^2 + (y + 6)^2 = 25$ represents the circle with center $(3,-6)$ and radius 5 units.

Circlecoordinate1.PNG

Circumference and Area

Given a circle of radius $r$, the circumference (distance around a circle) is $2 \pi r$ and the area is $\pi r^2$. Both formulas involve the mathematical constant pi ($\pi$).

Archimedes' Proof of Area

We shall explore two of the Greek mathematician Archimedes demonstrations of the area of a circle. The first is much more intuitive.

Archimedes envisioned cutting a circle up into many little wedges (think of slices of pizza). Then these wedges were placed side by side as shown below:

Pizzawedges2.PNG

As these slices are made infinitely thin, the little green arcs in the diagram will become the blue line and the figure will approach the shape of a rectangle with length $r$ and width $\pi r$ thus making its area $\pi r^2$.

Archimedes also came up with a brilliant proof of the area of a circle by using the proof technique of reductio ad absurdum.

Archimedes' actual claim was that a circle with radius $r$ and circumference $C$ had an area equivalent to the area of a right triangle with base $C$ and height $r$. First let the area of the circle be $A$ and the area of the triangle be $T$. We have three cases then.

Case 1: The circle's area is greater than the triangle's area.

Case 2: The triangle's area is greater than the circle's area.

Case 3: The circle's area is equal to the triangle's area.

Assume that $A>T$. Let $P$ be the area of a regular polygon that is closest to the circle's area. Therefore we have $A-P<A-T$ so $P>T$. Let the apothem be $a$ and the perimeter be $p$ so the area of a regular polygon is one half of the product of the perimeter and apothem. The perimeter is less than the circumference so $p<2\pi r$ and the apothem is less than the radius so $a<r$. Therefore $P=\frac{1}{2}ap<\frac{1}{2}r\cdot 2\pi r=T$. However it cannot be both $P>T$ and $P<T$. So $A\not >T$.

This article is a stub. Help us out by expanding it.

Area Proof Using Calculus

Let the circle in question be $x^2 + y^2 = r^2$, where r is the circle's radius. By symmetry, the circle's area is four times the area in the first quadrant. The area in the first quadrant can be computed using a definite integral from 0 to r of the function $f(x) = \sqrt{r^2 - x^2}$. Using the substitution $x = r \sin u, dx = r \cos u$ gives the indefinite integral as $\frac{r^2}{2} (u - \frac{\sin 2u}{2}) + C$, so the definite integral equals $\frac{r^2}{2} * \frac{\pi}{2}$. Multiplying by four gives the area of the circle as $\pi r^2$.

Lines in Circles

[asy]draw(unitcircle);draw((-0.8,1)--(1,1),Arrow);draw((1,1)--(-0.8,1),Arrow);draw((0,1)--(1,0));[/asy]

A line that touches a circle at only one point is called the tangent of that circle. Note that any point on a circle can have only one tangent.

A line segment that has endpoints on the circle is called the chord of the circle. If the chord is extended to a line, that line is called a secant of the circle. The longest chord of the circle is the diameter; it passes through the center of the circle.

When two secants intersect on the circle, they form an inscribed angle.

Properties

  • The measure of an inscribed angle is always half the measure of the central angle with the same endpoints.
    • Since the diameter divides the circle into two equal parts, any angle formed by the two endpoints of a diameter and a third distinct point on the circle as the vertex is a right angle.
    • Also, a right triangle inscribed in a circle has a hypotenuse that is a diameter of the circle.
  • Similarly, if a tangent line and a secant line intersects at the point of tangency, the measure of the angle formed is always half the measure of the central angle with the same endpoints.
    • From that property, the angle formed by the diameter and a tangent line with the point of tangency on the diameter is a right angle.
    • The perpendicular line through the tangent where it touches the circle is a diameter of the circle.
  • The perpendicular bisector of a chord is always a diameter of the circle.
  • When two chords $AB$ and $CD$ intersect at point $P$ inside the circle, $\angle APC = \frac{m\widehat{AC} + m\widehat{BD}}{2}$.
  • When two chords $AB$ and $CD$ intersect at point $P$ outside the circle, $\angle APC = \frac{m\widehat{AC} - m\widehat{BD}}{2}$.
  • Lengths of chords can be calculated by using the Power of a point theorem.

Problems

Introductory

  • Under what constraints is the circumference (in inches) of a circle greater than its area (in square inches)?

Intermediate

\[\mathrm{(A) \ } 13\qquad\mathrm{(B) \ } \frac{44}{3}\qquad\mathrm{(C) \ } \sqrt{221}\qquad\mathrm{(D) \ } \sqrt{255}\qquad\mathrm{(E) \ } \frac{55}{3}\qquad\]

(Source)

  • Let

\[S_1=\{(x,y)|\log_{10}(1+x^2+y^2)\le 1+\log_{10}(x+y)\}\]

and

\[S_2=\{(x,y)|\log_{10}(2+x^2+y^2)\le 2+\log_{10}(x+y)\}\]. What is the ratio of the area of $S_2$ to the area of $S_1$?

\[\mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D) \ } 101\qquad \mathrm{(E) \ }  102\]

(Source)

Olympiad

(<url>viewtopic.php?=217167 Source</url>)

See Also

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