# Equation Of A Circle Formula

## Equation of a Circle Formula

The circle equation formula refers to the circle equation that describes the shape of the centre radius of the circle. A circle refers to a circular boundary where all points on the boundary are equidistant from the centre. An equation is usually needed to represent a circle, and there are basically two forms of representation.

• Standard shape
• general shape

Students in this article will learn to solve the problems using the Equation Of A Circle Formula. The definition of a circle is the set of all points on the plane that are a fixed distance from the centre. Students must be able to recognise and apply the equation of a circle centred at the origin.

Equation Of A Circle Formula: The circle equation provides an algebraic way to describe a circle given its centre of radius and length. The Equation Of A Circle Formula is different from the formula used to calculate the area or circumference of a circle. This equation is used for many circle problems in coordinate geometry. To represent a circle in the Cartesian plane, we need the equation of the circle. Once students know the centre and radius length, students can draw a circle on paper. Similarly, if students know the coordinates of the centre and its radius, students can draw a circle on the Cartesian plane. Circles can be represented in various forms.

• General shape
• Standard shape
• Parametric shape
• Pole shape

In this article, students will learn about the circle equation, different forms with graphs, and solved examples.

What is the Equation Of A Circle Formula in geometry?

The Equation Of A Circle Formula describes the locus of a point at a constant distance from a fixed point. This fixed point is known as the centre of the circle and the constant value is the radius of the circle and standard circular equation.

What is the Equation Of A Circle Formula with the centre as the origin? In the simplest case, the circle is centred at the origin (0, 0) and has a radius of r and (x, y) is any point on the circumference. The equation for a circle, when centred at the origin, is x2 + y2 = r2.

What is the parametric equation for a circle? What is C in the general equation of the circle? The general form of the is Equation Of A Circle Formula x2 + y2 + 2gx + 2fy + c = 0. This general form is used to find the coordinates of the centre and the radius of the circle, where c is the constant term and the c-valued expression represents a circle that will not pass from the origin.

What are the different forms of the Equation Of A Circle Formula? What is the equation of a circle when its centre is on the x-axis? (x, y) will be any point on the circumference. The equation for the circle is if the centre is on the x-axis.

How do students graph a circle’s equation? To draw an Equation Of A Circle Formula, first, use the circle’s equation to determine the coordinates of the circle’s centre and the circle’s radius. Then draw a centre point on the Cartesian plane, use a compass to measure the radius, and draw a circle.

• How do students find the general Equation Of A Circle Formula?
• How do students write the standard form of the Equation Of A Circle Formula?
• How do Students arrive at the general form from the standard form of the Equation Of A Circle Formula?
• How would students write the standard form of a circle equation with endpoints?

Suppose the two endpoints of the diameter are (1, 1) and (3, 3). First, calculate the midpoint using the section formula. The coordinates of the centre then will (2, 2). Then calculate the radius from the distance formula between (1, 1) and (2, 2). The radius is the same √2

Now, the Equation Of A Circle Formula is in normal form

(x−2)2+(j−)2=2.

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Mathematics develops the ability to think because finding a solution requires imagining a whole coherent process. Mathematics can be said to be the foundation of children’s education because it teaches children the ability to think.

Mathematics allows people to explain how things work. In other words, students can express their thoughts and ideas clearly, coherently, and accurately. This is basic and very positive. Everyone else understands us and knows that people have clear and coherent thinking. How they present themselves correctly is a big part of the picture. Mathematics is present in all life and applies to new technologies as well as other sciences. In fact, many phenomena in everyday life are governed by exact science. Mathematics classes help and empower students to reach their beliefs because they teach that solutions to problems must lead to truth. Because it is definitely objective and logical. Mathematics speeds up thinking and generally helps students think more deeply about complex problems. Most people’s lives consist of decisions, approaches, considerations, and situations that confront problems that need to be resolved. In this sense, mathematics helps us to open our minds and understand that there is only one way to solve things.

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## What is the Equation of Circle?

What is the Equation Of A Circle Formula? The circle equation describes the position of the circle in the Cartesian plane. If students know the coordinates of the centre of the circle and the length of the radius, we can write the equation of the circle. The circle equation describes all the points on the circumference of the circle.

A circle represents the locus of points at a constant distance from a fixed point. This fixed point is known as the centre of the circle, and the constant value will be the radius r of the circle.

### Different Forms of Equation of Circle

An Equation Of A Circle Formula describes the position of a circle on the Cartesian plane. A circle can be drawn on a piece of paper given a centre and a radius length. Once students have found the coordinates of the centre of the circle and its radius using the circle equation, students can draw the circle in the Cartesian plane. There are various forms of representing the Equation Of A Circle Formula.

Here students will explore two general forms of the Equation Of A Circle Formula, the general and normal forms of the circle equation, and the polar and parametric forms.

### General Equation of a Circle

The general form of the Equation Of A Circle Formula is x2 + y2 + 2gx + 2fy + c = 0. This general form of the Equation Of A Circle Formula  is used to find the coordinates of the centre and radius of a circle. Where g, f, and c are constants. In contrast to the easy-to-understand standard form, the general form of the circle equation makes it difficult to find meaningful properties for a particular circle. So use quadratic completion for quick conversion from general form to normal form.

### Standard Equation of a Circle

The standard Equation Of A Circle Formula provides accurate information about the centre and radius of a circle, making it easy to read the centre and radius of a circle at a glance.

### Parametric Equation of a Circle

The general form of the Equation Of A Circle Formula is x2 + y2 + 2hx + 2ky + C = 0. Take a general point at the edge of the circle, say (x, y). The line connecting this general point to the centre of the circle (-h, -k) is θ

The parametric equation for the circle can be written as x2 + y2 + 2hx + 2ky + C = 0. Where x = -h + rcosθ and y = -k + rsinθ.

### Polar Equation of a Circle

The polar form of the circular Equation Of A Circle Formula is almost identical to the parametric form of the circular equation. Usually, it is written in the polar form of the Equation Of A Circle Formula centred at the origin. Students have to take a point P(rcosθ, rsinθ) on the boundary of the circle where r is the distance of the point from the origin. We know that the Equation Of A Circle Formula centred at the origin and radius ‘p’ is x2 + y2 = p2.

Substitute the x = rcosθ and y = rsinθ values ​​into the Equation Of A Circle Formula.

(rcosθ)2 + (rsinθ)2 = p2

r2cos2θ + r2sin2θ = p2

r2(cos2θ + sin2θ) = p2

r2(1) = p2

r = p

where p is the radius of the circle.

Example: Given the Equation Of A Circle Formula in standard form is x2 + y2 = 9, find the equation of the circle in polar form.

Resolution:

To find the Equation Of A Circle Formula in polar form, substitute the following values:

When:

x = rcosθ

y = rsin θ

x = rcosθ

y = r sin θ

x2 + y2 = 9

(rcosθ)2 + (rsinθ)2 = 9

r2cos2θ + r2sin2θ = 9

r2(cos2θ + sin2θ) = 9

r2(1) = 9

r = 3

### Equation of a Circle Formula

Circular equation calculations use circular equation formulas. By applying the Equation Of A Circle Formula students can find the equation of any circle, given the coordinates of the circle’s centre and its radius. Is the centre of a circle of radius r and (x, y) any point on the circumference?

Circle Terms

• Centre- The centre of the circle is a fixed point there.
• Radius- The radius of a circle is the specified distance from the center to the outermost point.
• Diameter- The diameter of a circle is the line segment that connects his two endpoints of the circle and crosses the center of the circle. A circle has a diameter which is twice its radius.
• Bow- An arc is a path that connects two points on the ends of a circle. A small arc is indicated by a small distance between two locations, and a large arc is indicated by a large distance.
• Chord- A chord is a portion of a straight line connecting two points at the ends of a circle.
• Sector- A sector is an area formed by connecting the ends of an arc to its center. The major sector is the larger area created and the minor sector is the smaller area formed between the arc and her two radii.
• Segment- A segment is an area created by connecting the ends of an arc using a chord. The major segment between the strings and the bow is the major segment, and the minor part is the minor segment.
• Secant- A secant is a line tangent to both endpoints of a circle.
• Tangent- A tangent is a line that touches a circle only once.

Circular object example

1. Edible

Cookies, cakes, doughnuts, pancakes, pizza, and many other foods are circular. The next time students chew any of these foods, remember the importance of the terminology associated with geometric circles.

2. Vinyl record

Records with modulated grooves are known as vinyl records or records. used for playing and storing audio data. The shape of the record is round. As a result, it is one of the largest examples of circular objects used in everyday life.

3. Hula Hoop

Hula hoops are toys that people twist around their arms, limbs, or waist for entertainment and physical health. The round shape of hula hoop tires is clearly visible.

4. Coin

The structure of the coin is perfectly spherical and rounded. They are remarkable examples of circular geometric shapes found in nature.

5. Ornaments

Most of the jewellery we wear has a round shape. Rings, bracelets, earrings, bangles, and other jewellery are all ideal examples of circular shapes.

6. Cooking

Most serving platters are round. Therefore, the most typical examples of circular objects used in everyday life are bowls

Meaning of circles in real life

The circle is still symbolically important today. They often represent peace and unity. For example, look at the Olympic logo. It contains five overlapping rings of different colours, symbolizing the world’s five major continents united in a spirit of healthy competition.

Conclusion

A circle is used from the nib to the shape of the planet. In life, circles exist all around us, but we never see them, even when they are right in front of us. A circle will appear like this. The diameter of a circle is the line segment that connects the two endpoints of the circle and crosses the centre of the circle.

### Derivation of Circle Equation

Derivation of Equation Of A Circle Formula given that is the centre of a circle of radius r and (x, y) is any point on the circumference. The distance from this point to the centre is equal to the radius of the circle. Now let’s apply the distance formula between these points.

Example: Use the Equation Of A Circle Formula to find the centre and radius of a circle whose equation is (x – 1)2 + (y + 2)2 = 9.

Resolution:

Answer: The circle is centred at (1, -2) and has a radius of 3.

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### Graphing the Equation of Circle

To show how the Equation Of A Circle Formula works, let’s graph a circle using the equation (x – 3)2 + (y – 2)2 = 9. Now, before graphing this equation, we need to make sure that the given equation is in canonical form.

So the circle represented by the formula (x – 3)2 + (y – 2)2 = 32 is centred at (3, 2) and has a radius of 3. The image below shows the graph obtained from this Equation Of A Circle Formula.

There are different ways of representing the Equation Of A Circle Formula, depending on the position of the circle on the Cartesian plane. Students have studied the geometry that represents the circle equation for the given coordinates of the centre of the circle. There are certain special cases depending on the position of the circle in the coordinate plane. Learn how to find the equation of the circle for the general case and this case.= 0 Answer: The equation of a circle when centred at the origin is x2+y2=r2.

• Equation of circle centred on the x-axis

Consider the case where the centre of the circle is on the x-axis: (a, 0) is the centre of the circle of radius r and (x, y) is any point on the circumference. The distance from this point to the centre is equal to the radius of the circle.

• Equation of circle tangent to both axes

Consider a circle whose circumference is tangent to both axes at one point: (r, r) is the centre of a circle of radius r. If the circle is tangent to both the x-axis and the y-axis, both coordinates of the circle’s centre are equal to the radius (r, r) and (x, y) is any point on the circumference. The distance from this point to the centre is equal to the radius of the circle. If the circle is tangent to both axes, we consider the centre of the circle to be (r, r). Where r can be the radius of the circle. (r, r) can be either positive or negative. Example: If the circle has a radius of 3 and is tangent to both axes, the centre coordinates are (3,3), (3,−3), (−3,3), or (−3,−3) will be.

• Conversion from general form to normal form

This is the standard equation for a circle centred at (a, b) with radius r: (x – a)2 + (y – b)2 = r2 and has the general form Consider: x2 + y2 + 2gx + 2fy + c = 0 The steps to convert general form to normal form are:

Step 1: Combine the same terms to get the opposite constant as x2 + 2gx + y2 + 2fy = – c → (1).

Step 2: Use the full quadratic identity (x + g)2 = x2 + 2gx + g2 to find the values ​​of the equations x2 + 2gx and y2 + 2fy as

(x + g)2 = x2 + 2gx + g2 ⇒ x2 + 2gx = (x + g)2 – g2 -> (2)

(y + f)2 = y2 + 2fy + f2 ⇒ y2 + 2fy = (y + f)2 – f2 -> (3)

Substituting (2) and (3) into (1), the equation becomes

(x+g)2 – g2 + (y+f)2 – f2 = -c

(x+g)2 + (y+f)2 = g2 + f2 – c

Comparing this expression with the standard form (x – a)2 + (y – b)2 = r2,

Before applying the formula, we need to make sure that the coefficients of x2 and y2 are 1.

Consider the example of finding the centre and radius of a circle from the general circle equation x2 + y2 – 6x – 8y + 9 = 0.

The coordinates of the centre of the circle will be found  (-g,-f). Where g = -6/2 = -3 and f = -8/2 = -4. So the centre is (3,4).

Therefore, the radius r = 4.

• Conversion from normal form to general form

The algebraic identity (a – b)2 = a2 + b2 – 2ab can be used to convert the standard form of

Notes on the Equation Of A Circle Formula:

Here are some points to keep in mind when learning the Equation Of A Circle Formula.

The general form of the Equation Of A Circle Formula always begins with x2 + y2. If the circle intersects both axes, there are four points of intersection between the circle and the axes. If the circle is tangent to both axes, there are only two tangent points. The radius is the distance from the centre to any point on the boundary of the circle. Therefore, the circle radius value is always positive.

### Examples on Circle Equations

Example 1: Find the standard form Equation Of A Circle Formula for a circle with centre (2,-3) and radius 3.

The standard form of the Equation Of A Circle Formula is written as

(2, -3) is the centre of the circle with radius r = 3.

Let’s put these values ​​into the standard form of the Equation Of A Circle Formula.

(x – 2)2 + (y – (-3))2 = (3)2

(x – 2)2 + (y + 3)2 = 9 is the required standard form of the given circle equation.

Example 2: Write the Equation Of A Circle Formula in standard form for a circle with centre (-1, 2) and radius of 7.

The standard form of the Equation Of A Circle Formula is written as

(-1, 2) is the centre of the circle with radius r = 7.

Let’s put these values ​​into the standard form of the Equation Of A Circle Formula

(x – (-1))2 + (y – 2)2 = 72

(x + 1)2 + (y – 2)2 = 49 is the required standard form of the given Equation Of A Circle Formula.

Example 3: Given the standard form Equation Of A Circle Formula  x2 + y2 = 16, find the polar form circle equation.

To find the Equation Of A Circle Formula in polar form, replace the x and y values ​​as follows:

x = rcosθ

y = r sin θ

x2 + y2 = 16

(rcosθ)2 + (rsinθ)2 = 16

r2cos2θ + r2sin2θ = 16

r2(1) = 4

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### Practice Questions on Equation of Circle

1. How do students derive the equation of the circle?

Answer: The centre coordinates of the circle are represented by (a, b), the radius is represented by r, and (h, k) is the circle of the circle Any point on the circumference. The distance between any point and the centre of the circle is equal to the radius of the circle. From the distance formula

√(h-a)²+(k-b)²=r

Squaring both sides gives

(h – a)² + (k – b)² = r², which is the standard equation for a circle. 2. What is the equation of the circle centred at the origin (0,0)?

Substituting the values ​​of (a, b),

(h – 0)² + (k – 0)² = r²

h² + k² = r²

The equation of a circle centred at the origin, where (h, k) is any point on the circumference of the circle.

1. Are there different forms of the circle equation?

Answer: The different forms of the circle equation are:

General shape:

x2+y2+2gx+2fy+c=0

Standard form: (h – a)²+ (k – b)² = r² where (h, k) is any point on the circumference and (a, b) are the centre coordinates. Polar Form: r=A is the polar form of the circle equation.

2. What is the polar form of the circle equation?

Answer: Substitute the values ​​of x = r cos θ and y = r sin θ into x2 + y2 = a2.

(r cos θ)2+ (r sin θ)2 = a2

r2cos2 θ + r2sin2 θ = a2

r2(cos2θ+sin2θ)=a2

r2(1) = a2 (since cos²θ + sin²θ = 1 from the trigonometric identity)

r = a

## 1. What is the Equation Of A Circle Formula according to geometry?

The Equation Of A Circle Formula describes the locus of a point at a constant distance from a fixed point. This fixed point is known as the centre of the circle, and the constant value is the radius of the circle.

## 2. What is the Equation Of A Circle Formula with the centre as the origin?

In the simplest case, the circle is centred at the origin (0, 0) and has a radius r and (x, y) is any point on the circumference. The equation for a circle when centred at the origin is x2 + y2 = r2.

## 3. What is the parametric equation for a circle?

What is C in the general equation of the circle? The general form of the Equation Of A Circle Formula is x2 + y2 + 2gx + 2fy + c = 0 and this general form will be used to find the coordinates of the centre of the circle and also the radius of the circle where c is the constant term and the c-valued expression represents a circle that does not pass through the origin.

## 4. How do students graph an Equation Of A Circle Formula?

To draw a circle’s equation, first use the Equation Of A Circle Formula to determine the coordinates of the circle’s centre and the circle’s radius. Then draw a centre point on the Cartesian plane, use a compass to measure the radius, and draw a circle.

## 5. Where can one find the study material for the Equation Of A Circle Formula?

One can find the study material and learning resources on the topic of Equation Of A Circle Formula on the Extramarks website or mobile application. Students also have the provision to download the required study notes in PDF format from the website.