A sector of a circle radius 6cm subtend an angle of 105° at the center of the circle calculate
I. The perimeter of the sector
Ii. The area of the sector
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.