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Arc

 

Do you know how an arc is formed? Well, an arc is a part of a circle. If we cut the circle from one end to another, we will get an arc. That end may or may not pass through the centre of the circle. Strictly speaking, an arc is a part of the circumference of the circle. The circumference is the boundary of the circle, also known as the perimeter.

Mathematically, an arc is referred to as the smooth curve that joins two points separated at a distance. An arc can be a part of a circle, an ellipse or even a hyperbola. It is sometimes referred to as the segment of the circle as well.

If we extend the ends of an arc, it will form a complete circle. In Euclidean geometry, the symbol of arc is given by ⌒ or ⌢. It is also denoted as and is pronounced as arc AB.

Arc length

Arc length is the space between two points along the curve. An angle is formed if we extend the ends of an arc joining the two endpoints along a straight line. In the given image, the angle OQP is the angle of the arc, and L is the arc length.

Arc length formula

The length of an arc can be calculated using the central angle subtended by the arc. The arc length formula is given by: L = θ x r where, L is the length of the arc w is the radius of the circle θ is the angle subtended at the centre of the circle in radians. 

How to find the angle subtended at the centre of the circle? From the length of the arc formula, we have, L = θ x r Therefore,

θ = L x r

This value will come in radians. To convert radians to degrees, we need to multiply by π/180. Therefore, θ (in radians) x π/180 = θ (in degrees)

Example 1:

Find the length of an arc if the angle formed by it is π/4 in a circle, given that the radius is equal to 6 units.

Solution:

From the arc length formula, we know, L = θ x r We have, θ = π/4, r = 6 Therefore, L = π/4 x 6 = 3/2 π units.

Example 2:

What is the length of an arc if the angle subtended by the circle is 75 degrees with a radius of 4 cm?

Solution:

From the arc length formula, we have, L = θ x r Here, the angle is in radians. To convert degrees into radians, we need to multiply the radian angle by π/180. Therefore, we get, 75 degrees = 75 x π/180 = 25 π/36 radians. From the formula we have, L = 25 π/36 x 4 = 25 π / 9 cm.

Applications of an arc

 1. Arc is most widely used in the construction of bridges over streams and rivers.
 2. They are used to construct arches over a roof or form the shape of a dome.
 3. It can be visualized easily and gives the building an aesthetic appearance.
 4. The figure is also used to transfer TV and radio signals with a high-speed cable.

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