An Arc of length 6 cm subtends a 80 degree angle in a circle. What is the radius of the circle? Also what is the area of that sector?
What I have so far:
S=R*80 degree
S= R*4pi/9
6 cm= r* 4pi/9
R=6/4pi/9
R= 6*9/4pi
R=54/4pi
HELP!
r=6/(4PI/9)=54/(4PI)=27/(2PI)
area of sector=PI r^2(360/80)=
= PI(6^2)(9/2)=PI18*9 check that.
Answer the following
Find the arc length and area of a sector and a circle if the arc measure is 80• and the length of the radius is 6 cm.
To find the radius of the circle, you can use the formula for the length of an arc, which is given by:
S = Rθ
Where S is the length of the arc, R is the radius of the circle, and θ is the angle (in radians) subtended by the arc.
In this case, you are given that the length of the arc is 6 cm and the angle is 80 degrees. However, since the formula for the length of an arc requires the angle to be in radians, you need to convert 80 degrees to radians.
To convert degrees to radians, you can use the formula:
θ (rad) = (π / 180) * θ (degrees)
Therefore, for 80 degrees:
θ (rad) = (π / 180) * 80 = (4/9)π
Now, substitute this value of θ in the formula S = Rθ and solve for R:
6 cm = R * (4/9)π
To isolate R, divide both sides of the equation by (4/9)π:
R = (6 cm) / ((4/9)π)
Simplify the expression by multiplying the numerator and denominator by 9/4:
R = (6 cm) * (9/4) / π
Now, calculate the value of R:
R = (54 cm) / π
So, the radius of the circle is 54/π cm.
To calculate the area of the sector, you can use the formula:
A = (θ/2) * R²
Where A is the area of the sector, θ is the angle (in radians) subtended by the sector, and R is the radius of the circle.
In this case, you already have the value of θ (80 degrees converted to radians) and the radius R (54/π cm). Substitute these values into the formula and calculate:
A = ((4/9)π/2) * (54/π)²
Simplify the expression:
A = (2/9) * (54/π)²
A = (2*54²) / (9π)
A ≈ 162 / π cm²
So, the area of the sector is approximately 162/π cm².