The angle of the arc as a fraction of a whole circle (360 degrees), multiplied by the circumference of a circle (\(2\pi r\)), gives the arc length.

Using the arc length formula:

\(l=\frac { 30° }{ 360° } \times 2\times \pi \times 5=2.62\quad cm\)

Using the formula for the area of a sector:

\(50=\frac { 45° }{ 360° } \times \pi { r }^{ 2 }\)

Dividing both sides by \(\frac { 45° }{ 360° }\) and by \(\pi\):

\({r}^2=127.324...\)

\(\therefore \quad r=11.3cm\)

We could use the formula for the area of a sector, with the angle, \(\theta\), being 180 degrees. However, it may be more efficient to use the formula for the area of a circle and then divide that area by 2 to get the area of the semi-circle.

\(A=\frac { \pi \times { 10 }^{ 2 } }{ 2 } =157.0796{ cm }^{ 2 }=157.1{ cm }^{ 2 }\)

Using the formula for the arc length of a sector:

\(A=\frac { 120° }{ 360° } \times 2\times \pi \times 6=\frac { 1 }{ 3 } \times 12\pi =\frac { 12\pi }{ 3 } =4\pi \) cm

There are a number of ways to do this, one way being to find the interior angle of the sector, which is 320°, and then utilize the arc length formula to obtain 39.0954 cm. To then find the total perimeter, we must add 14 cm for the two radii, giving 53.1 cm to 3 significant figures.

The area of the sector is given by:

\(A=\frac { 10° }{ 360° } \times \pi \times { 8 }^{ 2 }=5.5851{ cm }^{ 2 }=5.59{cm}^2\)

Using the area of a sector formula:

\(A=\frac { 160° }{ 360° } \times \pi \times { 4 }^{ 2 }=22.3{ m }^{ 2 }\)

See answers.

Given that \(\theta\) is divided by 360°, we must multiply both sides by 360°. Also, since we are multiplying \(\theta\) by \(\pi{r}^2\), we must divide both sides by \(\pi{r}^2\). Essentially we are conducting the inverse operations when rearranging.