Question

How do you calculate the length of an arc and area of a sector?

Answer 1

Hint: The length of an arc of any circle depends on the radius of that circle and the central angle \[\theta \] (in radians). We know that for the angle equal to 360 degrees \[\left( 2\pi \right)\] , the arc length is equal to circumference. Similarly the area of sector can be found using the area of the whole circle.

Complete step by step solution:
 Length of an arc of a circle

The proportion between angle and arc length is constant, we can say that:
\[\dfrac{L}{\theta }\] = \[\dfrac{C}{2\pi }\]
As circumference C = \[2\pi r\],
\[\Rightarrow \]\[\dfrac{L}{\theta }\] = \[\dfrac{2\pi r}{2\pi }\]

\[\Rightarrow \]\[\dfrac{L}{\theta }\] = \[r\text{ }\]
We find out the arc length formula when multiplying this equation by θ on both the sides:
\[L=r\times \theta \]
Hence, the arc length is equal to radius multiplied by the central angle (in radians).
Area of a sector of a circle,

We can find the area of a sector of a circle in a similar way as we found the length of an arc. We know that the area of the whole circle is equal to \[\pi r{}^\text{2}\]. From the proportions,
\[\dfrac{A}{\theta }\] = \[\dfrac{\pi r{}^\text{2}}{2\pi }\]
\[\Rightarrow \] \[\dfrac{A}{\theta }\] = \[\dfrac{r{}^\text{2}}{2}\]
We find out the Area of a sector of a circle formula when multiplying this equation by θ on both the sides.
The formula for the area of a sector is:
\[\Rightarrow \] A = \[\dfrac{r{}^\text{2}}{2}\times \theta \]


Note: These are the standard formulas, so we have to know how we obtained these formulas. The important point here is that the values \[\theta \] is in radians, so if we have the \[\theta \] value in degrees then we should multiply the obtained formulas with \[\dfrac{360}{2\pi }\] .