Arcs and sectors | Mathematico

One reason why radians are so convenient is that they allow us to calculate arc lengths and areas much more easily.

The circumference and area of the whole circle are $2 π r$ and $π r^{2},$ so in degrees, we have

ℓ = 2 π r \times \frac{θ in degrees}{360}

and

A = π r^{2} \times \frac{θ in degrees}{360}

However, in radians, this simplifies to

\begin{aligned} ℓ & = 2 π r \times \frac{θ in radians}{2 π} \\ = r θ \end{aligned}

and

\begin{aligned} A & = π r^{2} \times \frac{θ in radians}{2 π} \\ = \frac{1}{2} r^{2} θ \end{aligned}

Thus, we get two neat formulas:

\begin{aligned} ℓ & = r θ \\ A & = \frac{1}{2} r^{2} θ \end{aligned}

A circular sector of radius $3 \sqrt{5}$ is drawn below. The angle at the center is $\frac{8 π}{5}$ radians.

Sector