## Equation of the circumference II: general equation

A circumference with center C = (a, b) and radius r can be rewritten in light of the reduced equation as:

Developing the squares of the above mentioned equation we obtain:

x^2+y^2-2ax-2by+a^2+b^2-r^2=0

and doing the change A= -2a, \ \ B=-2b, \ \ C=a^2+b^2-r^2 in:

x^2+y^2-2ax-2by+a^2+b^2-r^2=0

the new equation is obtained:

This way we have found another analytical expression that defines the points of a circumference. This is the general equation of the circumference.

Let’s see how to determine the radius and the center of a circumference from the general equation.

We can do the following:

A=-2a, \ \ B=-2b, \ \ C=a^2+b^2-r^2

We isolate these expressions in terms of a, b and r. We have:

\displaystyle a=-\frac{A}{2} r^2=a^2+b^2-C=\Big(-\frac{A}{2}\Big)^2+\Big(-\frac{B}{2}\Big)^2-C=\frac{A^2+B^2-4C}{4}

And since we know that, in the limited expression, (a, b) is the center and r the radius, given a general equation:

the center of such a circumference is the point \displaystyle \Big(-\frac{A}{2},-\frac{B}{2}\Big) and the radius is \displaystyle r=\sqrt{\frac{A^2+B^2-4C}{4}}.

Let’s suppose that they give us the circumference:

then we see that it is centred at the point:

\displaystyle \Big(-\frac{A}{2},-\frac{B}{2}\Big)=\Big(-\frac{-2}{2},-\frac{4}{2}\Big)=(1,-2)

and has radius:

\displaystyle r=\sqrt{\frac{A^2+B^2-4C}{4}}=\sqrt{\frac{(-2)^2+4^2-4\cdot(- 4)}{4}}=

=\displaystyle\sqrt{\frac{4+16+16}{4}}=\sqrt{\frac{36}{4}}=\frac{6}{2}=3

Let’s now see the inverse process, that is to say:

Giving the general equation of the circumference that has, for example, radius 4 and center (-5, 6).

We write the reduced equation:

(x-a)^2+(y-b)^2=r^2 \Rightarrow (x+5)^2+(y-6)^2=4^2

developing the squares we have:

(x+5)^2+(y-6)^2=4^2 \Rightarrow x^2+10x+25+y^2-12y+36=16

If we rearrange it and add all the independent terms, we obtain the general equation of the above mentioned circumference, which is:

Let’s see what happens when the circumference is centred on the origin and we want to write its general equation:

Since (0, 0) is the center we have: a=0 and b=0 for which reason,

\left.{\begin{matrix} {0=a=-\frac{A}{2}} \\ {0=b=-\frac{B}{2}} \end{matrix}}\right \}\Longrightarrow{\left \{ {\begin{matrix} {A=0}\\{B=0}\end{matrix}}\right . }

So that in the general equation, only quadratic terms and independent terms will exist, that is to say:

Moving the constant term to the other side we obtain:

where we know that:

since the supposed center was (0, 0).

Note that for the circunference centered at the origin both equations are very similar.

Let’s see an example:

Circumference centred on the origin and radius 7.

Reduced equation: x^2+y^2=7^2

General equation: x^2+y^2+C=0 where C=-7^2 \Longrightarrow x^2+y^2-7^2=0

Summing up we have:

Considering the circumference: (x-a)^2+(y-b)^2=r^2

Then the center is the point of the plane (a,b) and the radius is r.

(x-8)^2+(y+3)^2=1 has center at (8,-3) and radius 1.

Considering the circumference: x^2+y^2+Ax+By+C=0

Then the center is at the point of the plane \displaystyle \Big(-\frac{A}{2},-\frac{B}{2}\Big) and the radius is \displaystyle r=\sqrt{\Big(\frac{A}{2}\Big)^2+\Big(\frac{B}{2}\Big)^2-C}

x^2+y^2+x-5y-2=0 has center \displaystyle \Big(\frac{-1}{2},\frac{5}{2}\Big) and radius

\displaystyle r=\sqrt{\Big(\frac{1}{2}\Big)^2+\Big(\frac{-5}{2}\Big)^2-(-2)}=\sqrt{\frac{1+25+8}{4}}=\sqrt{\frac{34}{4}}=\sqrt{\frac{17}{2}}

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## Circumference of a Circle Equation

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To simplify calculations the value of is rounded to 3.14. The diameter of a circle is twice its radius. Therefore, Circumference, C = 2 r The distance around a closed curve is called as the circumference of the circle. It is also defined as the length around the circle. The circumference of a circle is measured in linear units like inches or centimeters. Know More About What is the Volume of a Sphere Circumference of a Circle Examples Circumference of a Circle Formula

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The distance around a closed curve is called as the circumference of the circle. It is also defined as the length around the circle. The circumference of a circle is measured in linear units like inches or centimeters. Circumference of a Circle FormulaThe formula of circumference of a circle is given by the following formula,Circumference, C = 2 * * rwhere, r is the radius of the circle. and pi is a constant value and is equal to 3.14 The circumference of a circle can be calculated by its diameter using the following formula:Circumference, C = * D Where, value of is a constant and its value is approximately 3.14159265358979323846…. or 22/7 to be more precise and D is the diameter of the circle. To simplify calculations the value of is rounded to 3.14. The diameter of a circle is twice its radius. Therefore, Circumference, C = 2 rCircumference of a Circle ExamplesCircumference of a Circle EquationKnow More About What is the Volume of a SphereTutorvista.com Page No. :- 1/6 Below are the examples on circumference of a circle -Example 1:Find the circumference of a circle given that its radius is 10.Solution:The circumference is always multiply by 2 the length of the radius,FormulaCircumference of circle = 2 rCircumference = 2 x 10 x = 62.8Example 2:Find the circumference of a circle known that area of the circle is 153.86Solution:Step 1:Find the radius of the circle . r2 = 153.86r2 = r2 = 49Learn More Arc of a CircleTutorvista.com Page No. :- 2/6 r = 7Step 2:Then calculate the Circumference of circle,Circumference of circle = 2 r= 2 * 7 * = 43.96Therefore, the circumference of the circle is 14.Example 3:Find the circumference of a circle known that its radius is 16Solution:The circumference is always multiply by 2 the length of the radius,Circumference of circle = 2 rCircumference = 2 x 16 x = 100.48Tutorvista.com Page No. :- 3/6 The Pythagorean theorem is related to the study of sides of a right angled triangle. It is also called as pythagoras theorem. The pythagorean theorem states that, In a right triangle, (length of the hypotenuse)2 = {(1st side)2 + (2nd side)2}.In a right angled triangle, there are three sides: hypotenuse, perpendicular and base. The base and the perpendicular make an angle of 90 degree with eachother. So, according to pythagorean theorem:(Hypotenuse)2 = (Perpendicular)2 + (Base)2In the above figure1,c2 = a2 + b2Therefore, Hypotenuse (c) = (a2 + b2)From the above figure 2, ABC is a right angled triangle at angle C.From C put a perpendicular to AB at H.What is Pythagorean TheoremTutorvista.com Page No. :- 4/6 Now consider the two triangles ABC and ACH, these two triangles are similar to each other because of AA similarity. This is because both the triangle have a right angle and one common angle at A.So by these similarity,ac = ea and bc = dba2 = c*e and b2 = c*dSum the a2 and b2, we geta2 + b2 = c*e + c*da2 + b2 = c(e + d)a2 + b2 = c2 (since e + d = c)Hence Proved.Euclid Proof of Pythagorean TheoremAccording to Euclid, if the triangle had a right angle (90 degree), the area of the square formed with hypotenuse as the side will be equal to the sum of the area of the squares formed with the other two sides as the side of the squares.From the above figure 3, the sum of the area covered by the two small squares is equal to the area of the third square. Here, a2 is the area of the square ABDE, b2 is the area of the square BCFG and c2 is the area of the square ACHI.Read More About Perpendicular Lines DefinitionTutorvista.com Page No. :- 5/6 ThankYouTutorVista.comSlide 1Slide 2Slide 3Slide 4Slide 5Slide 6

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## Circumference of a Circle Equation

Published on

05-Mar-2016View

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The distance around a closed curve is called as the circumference of the circle. It is also defined as the length around the circle. The circumference of a circle is measured in linear units like inches or centimeters. Know More About Geometry Distance Formula Circumference of a Circle Formula

Transcript

Circumference of a Circle EquationKnow More About Geometry Distance FormulaThe distance around a closed curve is called as the circumference of the circle. It is also defined as the length around the circle. The circumference of a circle is measured in linear units like inches or centimeters.Circumference of a Circle FormulaThe formula of circumference of a circle is given by the following formula,Circumference, C = 2 * * rwhere, r is the radius of the circle. and pi is a constant value and is equal to 3.14The circumference of a circle can be calculated by its diameter using the following formula:Circumference, C = * DWhere, value of is a constant and its value is approximately 3.14159265358979323846…. or 22/7 to be more precise and D is the diameter of the circle. To simplify calculations the value of is rounded to 3.14. The diameter of a circle is twice its radius. Therefore, Circumference, C = 2 r Circumference of a Circle ExamplesBelow are the examples on circumference of a circle -Example 1:Find the circumference of a circle given that its radius is 10.Solution:The circumference is always multiply by 2 the length of the radius,FormulaCircumference of circle = 2 rCircumference = 2 x 10 x = 62.8Example 2:Find the circumference of a circle known that area of the circle is 153.86Solution:Step 1:Learn More What is Distance Formula Learn about distance formula here and understand the concept better with solved examples provided. Students can also use the online distance formula calculator and distance formula worksheet provided in the page.Let’s understand what is the distance formula? The length of a line segment AB, which joins A (x1, y1) and B (x2, y2) is given by,Distance Formula ProofLet A (x1, y1) and B (x2, y2) be two points in the plane.Let d = distance between the points A and B.Draw AL and BM perpendicular to x-axis (parallel to y-axis).Draw AC perpendicular to BM to cut BM at C.In the figure,What is Distance Formula OL = x1, OM = x2 [AC = LM = OM — OL = x2 — x1]MB = y2, MC = LA = y1 [CB = MB — MC = y2 — y1]From the right-angled DACB,Distance Formula ExamplesBelow are some examples based on distance formulaExample 1: Find the distance between the following pair of points: A (1,2) and B (4,5).Solution:Using the distance formula, we haveExample 2: Find the distance between places when the two coordinates (2, 4) and (4, 6)are given, using the distance formula.?Solution:(x1, y1)= (2, 4)(x2, y2) = (4, 6)Read More About Pyramid Definition ThankYouTutorVista.comSlide 1Slide 2Slide 3Slide 4Slide 5

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