finding arc length and area of a sector?
If I know radius is 16cm and central angle of 50° = 5pi/18 radians. I have to use the equation s = (angle)r² to find arc length. is 40pi/9 cm correct?
and for area = (1/2) r²(angle) I got 1280pi/9 what did I do wrong?
i meant to say s = (angle) r
- Anonymous1 decade agoFavorite Answer
Since you first mentioned degrees instead of radians, I will work with degrees. I should point out that if work with radians, you can save several steps, but degrees it is.
Let P be the perimeter of the circle.
P = 2 * π * r.............In this case that is:
P = 2 * π * 16cm = 32 π cm. I'm going to leave the measure cm out until the end so that the work with numbers is more obvious.
Fact: Arc length is not proportional to r^2, it is proportional to r.
So the answer to your problem is the ratio of the central angle to the 360 degrees of the full circle times the perimeter. So:
s = (50/360) * P = [(5 * P) / (36)] = [(5 * 32 * π) / 36] = ((160 * π) / 36)
Divide the numerator (160 * π) and the denominator (36) by 4 getting:
s = ((40 * π) / 9) cm........<<<<<
Note: it was the r^2 in the formula that misled you about the arc length.
For the area you want to compute 50/360 times the area of the circle.
A (of the defined part of the circle) = (50/360)*π*r^2 = (5/36)*π*(16)^2 = (5/36)*π*256 =
5 * (256/36) * π = 5 * (64/9) * π = (320/9) * π square cm =
[(320 * π) / 9] square cm......<<<<<
- z_o_r_r_oLv 61 decade ago
s = r * (angle in radians)
A = pi r² (angle in radians / (2 pi))