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 Circles

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Geometry is a study of shapes and its properties. Circle is one of the basic shapes which we come across quite frequently in real life. Tyre and coins are the common example of a circle.

A circle can be defined as a shape where each point is equidistant from a certain point known as the center of the circle:
Circle with Radius and Diameter

The distance of each point from the center is known as the radius of the circle. The distance between any two points on the edge of a circle is called an arc of the circle and a line drawn between any two points is known as a chord of the circle.

Radius

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Radius of a circle is the distance of each point on a circle from the center. It is usually denoted by r. Radius is half of the length of the longest chord in the circle, that is, the diameter. Hence, the radius of a circle,  $r$= $\frac{d}{2}$where d is the diameter of circle.
Circle with Diameter
From the above image we can see that AO = OB = r, radius of the circle. AB is the longest chord passing the center, that is, the diameter of the circle, d. AB = AO + OB. So we have d = r + r = 2r.

An arc of the circle is a part of the perimeter of the circle, which is taken between any two points on the circle. Central angle is the angle made at center by the lines joining the arc points and the center. To find the radius of the circle, when the arc length and the central angle is given the following formula is used.
Radius = $\frac{Arc\  length}{central\  angle}$Suppose that in a circle the central angle is 1.25 radians and the radius is 5 inches. Then, the arc length, s = 5 * 1.25 = 6.25 inches. If the central angle is given in degree it is needed to be changed in radian.

Diameter

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A chord of a circle is the line joining any two points of the circle. The diameter is the longest chord of the circle that passes through its center. The diameter of the circle will always be twice the length of the radius of the circle. The diameter of the circle will divide the area of the circle into two halves, and the circumference of the circle will also be divided into equal halves. The formula to get the diameter of a circle, d = 2 * r, where r is the radius of the circle. 

Circumference

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The circumference of a circle is the distance around the circle, that is, the perimeter of the circle. We can understand it as if we stand on one point of the circle, and walk around the whole circle to come back to that point the distance traveled is the circumference of the circle. The formula for the circumference of the circle is, C = $2\pi r$ = $\pi d$.

We can see that the circumference is directly proportional to the diameter of the circle. From here we can find the definition for $\pi $ also. $\pi$ = $\frac{C}{d}$.
Area of a circle, A = $\pi r^{2}$
r = radius of the circle
$\pi $ = 3.14
Area of the circle is the total size occupied by the circle. A circle is a two-dimensional entity which can be drawn on a plane. Suppose that the radius of a circle is given as 5cm. Then the area of the circle will be $\pi 5^{2}= 25\pi cm^{2}$. This formula to find the area of the circle is also known as the circle formula.

Examples

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Example 1: A rope is lying on the ground in such a way that it makes a circle with diameter 12 m. If Alice stands on the rope and walks on it to come back on the same point. What is the distance walked by her?

Solution: The distance traveled will be the circumference of the circle. As the diameter is 12, circumference will be $\pi d$. Hence, distance traveled = $12\pi $ meters.

Example 2: The diameter of a circle is given as 6 cm. Find its area.

Solution: Diameter = 2*radius.

Radius of the circle, r =  $\frac{6}{2}$. = 3 cm.

Area of the circle, A = $\pi r^{2}$ = $\pi 3^{2}$ = $9\pi$

Therefore the area of given circle is $9\pi$cm^{2}$ .

Example 3: The arc length between two points on a circle is found to be 2 cm. Find the radius if the central angle is given as 0.5 radians.

Solution: Radius = $\frac{Arc\ length}{ central\  angle}$

Radius, r = $\frac{2}{0.5}$ = 4.

Hence, the radius is 4 cm.