Arc of a Circle

ARC OF A CIRCLES 😎

An arc is a portion of the total circumference of a circle. The arc of a circle is denoted by the symbol "⌢$⌢$".


           

Arc of a Circle Equation

              

In the figure, PQ is an arc of the circle with the same degree measure as central angle POQ. The shorter distance between P and Q (drawn in red) and longer distance between P and Q (drawn in blue) along the circle is called the minor arc and major arc respectively.

Arc Length of a Circle

In the above figure, the part of the circle L is called the arc of the circle.

Arc Length of a Circle Formula

The length of an arc of a circle with radius r and subtending an angle θ$\theta$ in the center of the circle is defined as,

• If θ$\theta$ is in radians, L = θ$\theta$ * r
• If θ$\theta$ is in degrees, L = θ360o$\frac{\theta }{{360}^o}$ * 2 π$\pi$ r

This is the formula for finding length of the arc of a circle. This is based on the principle that the total angle at the centre is 360 degrees. So, the arc subtending an angle θ$\theta$ will be θ360o$\frac{\theta }{{360}^o}$ * 2 π$\pi$ r. This can be used to find the length of the arc of a circle. The length of the arc to the perimeter is in the same proportion as the subtended angle to the centre to 2π$\pi$ radians.

For an arc with a central angle measuring θ$\theta$ degrees:

Arc Length = θ360$\frac{\theta }{360}$ * (Circumference of circle)

Where Circumference = 2π$\pi$ r (r = radius of a circle).

Area of Arc of Circle

To determine the area of a sector bounded by 2 radii and an arc in the circle, use the same method used to find the length of an arc.

                                                   

In a sector whose central angle measures is θ$\theta$ degrees.

Area of sector = θ360$\frac{\theta }{360}$ * Area of circle

Where, Area of circle = π$\pi$ r2$r^2$ (r = radius of a circle).

Let us find the area of the sector of the circle with 60 at the centre and radius 7.

Area of the sector = 60360$\frac{60}{360}$ * (227$\frac{22}{7}$ * 7 * 7) 

= 1546$\frac{154}{6}$

= 773$\frac{77}{3}$.

Move to next topic ➡ Area of a Circle