Circle Formulas

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Area of a Circle

The area of a circle can be considered as the number of square units of space the circle occupies.

For example, if you draw a square of 2 cm by cm inside the circle, then the total number of squares placed inside the circle represents the area of a circle. The units in which the area of the circle can be measured are m², km², in², mm², etc.


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Let us learn important circle formulas through this article.


Area of a Circle Formulas

The area of circle formula in terms of the radius is given as:


Area of a Circle = \[\pi r^{2}\]


The area of circle formula in terms of diameter is given as:


Area of a Circle = \[\frac{\pi}{4} \times d^{2}\]


In the above area of circle formulas, ris the radius and d of the circle.

The value of \[\pi\] is \[\frac{22}{7}\] or 3.14.


Surface Area of Circle Formula

The surface area of a circle is defined as the total area occupied by the circle. The surface area of circle formula is given as:


Surface Area of a Circle = \[\pi r^{2}\]


What is a Circle?

A circle is defined as the set of all points in the plane that maintains a fixed finite distance (r) from a fixed centre point (x,y). Here, r is the radius and O is the centre point of a circle.


Equation of a Circle

The equation of a circle has two forms. These are:

  1. The general form of the circle.

  2. The standard form of the circle.

If the equation of a circle is given in the standard form, we can easily identify the coordinates of a centre and radius (r) of a circle. It should be noted that the radius (r) is always positive.


General Equation of a Circle

The general equation of a circle with coordinates of a centre (h,k), and radius r is given as:


General Equation of a Circle = \[\sqrt{(x - h)^{2} + (y - k)^{2}} = r\]


Standard Equation of a Circle

The standard equation of a circle provides appropriate information about the centre point and radius of a circle and is, therefore, can be written and read easily.


The standard equation of a circle with centre (a,b), and radius r is given as:


Standard Equation of a Circle = \[(x - a)^{2} + (y - b)^{2} = r^{2}\]


A unique circle with centre point O = (a, b), and radius r can be constructed as shown below:


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Radius of a Circle Formula

The radius of a circle is the length of a line segment from the centre point of a circle to any point on its circumference. It is represented by “r”.


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The radius of a circle formula in terms of diameter is given as:


Radius of a Circle = Diameter/2


The radius of a circle formula in terms of the circumference is given as:


Radius of a Circle = Circumference/2π


If the area of a circle is represented by A, then the radius of a circle formula in terms of area is given as:


Radius of a Circle = \[\sqrt{\frac{A}{\pi}}\]


Diameter of a Circle Formula

The diameter of a circle is defined as the length of a line that starts from one point on the circle, passes through the centre, and ends on the other point on the circle's opposite side.


The diameter of a circle formula is given as :


Diameter of a Circle = 2 x radius


Circumference of Circle Formula

The circumference of a circle is the boundary or length of the complete arc of a circle.


The circumference of a circle formula is given as:


Circumference of a Circle = 2 × π × r


In the circumference of a circle formula, ris the radius of the circle and the value of π is 22/7 or 3.14.


Solved Examples

1. Find the area, circumference, and diameter of a circle of radius 5 cm.

Solution:

Area of a circle = π × r²

Here, r = 5 cm, π = 3.14

Area of a circle = 3.14 × 5² = 78.5 cm

Circumference of a circle = 2πr

Here, r = 5 cm, π = 3.14

Circumference of a circle = 2 × 3.14 × 5 = 31.4 cm

Diameter of a circle = 2 × radius

Here, r = 5 cm,

Diameter of a circle = 2 × 5 = 10 cm


2. What will be the equation of the circle whose centre is (2,6) and the radius is 4 units.

Solution:

Here, the centre of the circle is not an origin. Hence, the general equation of the circle will be applied.

The general equation of a circle = (x – x₁)² + (y – y₁)² = r²

Substituting the values:

(x - 2)² + (y - 6)²  = 4²

Hence, the required equation of a circle is (x - 2)² + (y - 6)²  = 4²


Conclusion

With these circle formulas, you will be easily able to solve different questions based on the equation of a circle, area of a circle, the diameter of a circle, the radius of a circle, and circumference. So, understand the concepts behind the formula and apply them wherever required.

FAQ (Frequently Asked Questions)

Q1. How to Calculate the Distance Around the Boundary of a Circle?

Ans. The distance around the boundary of a circle can be calculated using the circumference of a circle formula.

Q2. What are the Real-Life Examples of a Circle?

Ans. Two typical examples of a circle that you find in real -life is:

  1. Ferris Wheel - Each point located along the outer rim of the wheel is equidistant from the centre.

  2. Bicycle Wheel - Bicycle wheels are the best example of a circle. The circle shape is considered to be a feasible shape for a bicycle as they roll very easily because they are round in shape.

Q3. What are the Two Different Forms of an Equation of a Circle?

Ans. Two different forms of an equation of a circle are:

  1. The general form 

  2. The standard form