A sector of a circle radius 6cm subtend an angle of 105° at the center of the circle calculate

Question

I. The perimeter of the sector
Ii. The area of the sector

Answer 1

To find the perimeter of the sector, we need to know the length of the arc. The length of an arc can be calculated by using the formula:

\(Length \ of \ arc = \frac{2\pi r \times \theta}{360}\)

where r is the radius of the circle and θ is the central angle in degrees.

Given that the radius is 6 cm and the central angle is 105°, we can substitute these values into the formula:

\(Length \ of \ arc = \frac{2\pi \times 6 \times 105}{360}\)

Simplifying the equation:

\(Length \ of \ arc = \frac{2\pi \times 6 \times 7}{12}\)

\(Length \ of \ arc = 2\pi cm\)

So, the length of the arc is 2π cm.

Since the perimeter of the sector is made up of the arc length and two radii, we can calculate it as:

\(Perimeter \ of \ sector = 2\pi + 2r\)

Substituting the values:

\(Perimeter \ of \ sector = 2\pi + 2\times 6\)

\(Perimeter \ of \ sector = 2(\pi + 6)\)

Therefore, the perimeter of the sector is 2(π + 6) cm.

To find the area of the sector, we can use the formula:

\(Area \ of \ sector = \frac{\pi r^2 \times \theta}{360}\)

Using the given values:

\(Area \ of \ sector = \frac{\pi \times 6^2 \times 105}{360}\)

Simplifying the equation:

\(Area \ of \ sector = \frac{\pi \times 36 \times 7}{12}\)

\(Area \ of \ sector = 14\pi \ cm^2\)

Therefore, the area of the sector is 14π cm².