**What is a circle?**

A circle is the set of all those points in a plane whose distance from the fixed point remains constant. The fixed point is called the **centre** of the circle and the constant distance is known as the **radius** of the circle. The line segment passing through the centre of a circle and having its end points on the circle is called the **diameter** of the circle.

**What is an arc in a circle?**

An arc of a circle is referred to as a curve that is a part or portion of its circumference. Acute central angles will always produce minor arcs and small sectors. When the central angle formed by the two radii is 90^{o}, the sector is called a quadrant because the total circle comprises four quadrants or fourths. When the two radii form a 180^{o}, or half the circle, the sector is called a semicircle and has a major arc.

Other important components of a circle include –

**Segment in a Circle: **The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.

**Sector of a Circle: **The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors, minor sector, and major sector.

**What are radians?**

Radians are another way of measuring angles, and the measure of an angle can be converted between degrees and radians. Some key points regarding radians are –

- One radian is the measure of the central angle of a circle such that the length of the arc is equal to the radius of the circle.
- One complete circle which measures 360
^{o}is equal to 2π radians. This implies that 1 radian = $\frac{180^o}{\pi}$ - The formula used to convert between radians and degrees is defined by –

**Angle in degrees = Angle in radians x **$\frac{180^o}{\pi}$

- If s is the length of an arc of a circle, and r is the radius of the circle, then the measure of the radian is given by the central angle that contains that arc.

**Example**

Convert an angle measuring $\frac{\pi}{9}$ radians to degrees.

**Solution**

In order to convert the given radians into degrees, we will use the relation between the radians and the degrees that we have defined above. We know that –

Angle in degrees = Angle in radians x $\frac{180^o}{\pi}$

So, Angle in degrees = $\frac{\pi}{9} x \frac{180^o}{\pi}$ = 20^{o}

** Hence, **$\frac{\pi}{9}$** radians = 20 ^{o}**

**How to find the length of an arc of a circle?**

The term, length of the arc means the measure of the distance along the curved line making up the arc. This length can be calculated both in radians as well as degrees. Below we have the formula for finding the length of arc when provided with various other dimensions –

Let “ s “ be the length of the arc of a circle, “ r “ be the radius of the circle and the constant π = $\frac{22}{7}$ . Then,

**Arc Length Formula ( if θ is in degrees ) ( s ) = 2 π r ( **$\frac{θ}{360^o}$** ) **

**Arc Length Formula ( if θ is in radians ) ( s ) = ****r**

Let us understand it using an example

**Example**

Calculate the length of an arc if the radius of an arc is 8 cm and the central angle is 40°.

**Solution**

We have been given that

Radius ( r ) = 8 cm

Central angle ( ) = 40°

Now, we know that

Arc Length Formula ( if θ is in degrees ) ( s ) = 2 π r ( $\frac{θ}{360^o}$ )

Substituting the given values in the above equation, we will have,

Arc Length = 2 x π x 8 x ( $\frac{40^o}{360^o}$ ) = 5.582 cm

**Hence, the length of the arc if the radius of an arc is 8 cm and the central angle is 40° = 5.582 cm.**

**Using the arc length calculator for finding the length of an arc of a circle**

It is quite simple to use the scientific notation calculator for performing operations involving scientific notations. The following steps are required to be followed for this purpose –

**Step 1** – The first step is the know how of what the arc length calculator looks like.** **The following is the snapshot of what the landing page would look like as soon as click on the “arc length calculator“ link –

**Step 2** – as we can see in the snapshot above, the calculator needs some information to be provided as input for the purpose of calculations. These are the radius of the circle and the angle in radians. Therefore, in the second step, we will first enter the radius of the circle. Let us take, for example, the radius of the circle as 2 cm. therefore we will enter 2 in the box provided for “ radius “ in the “ Enter Information “ section. Below is the snapshot of how the radius shall be entered –

**Step 3** – The next step is to enter the angle in radians. Let us for example, take the angle as 2 radians. So we now have the situation where we are trying to find the length of the arc of a circle with a radius of 2 cm and central angle as 2 radians. We will now enter “ angle in radians “ as 2 in the “ Enter Information “ section. Below is the snapshot of how the angle in radian shall be entered –

**Step 4** – Now that we have entered all the required information, our last step is to perform the calculation. For this purpose, we just need to click on the “ calculate “ button. As soon as we will click on this button, we can see the result obtained on the right-hand side of the values that we had entered in the previous steps. Below is a snapshot of how the selection would look like when we will click on the “ calculate “ button –

We can clearly see in the screenshot above that the calculator provides us with not just the values of the arc length, but also gives us the sector area, the length of the chord as well as the length of the diameter. All the associated calculations are provided for reference and understanding purposes. Hence, repeated use of this arc length calculator shall help understand more about degrees, radians, arc length, sector area and other related concepts in circles.