sectors of circles and apparent size

K = area of the sector
r = radius
s = the arc length
Θ = the central angle always in radians

FORMULAS
K = ½r²Θ
K = ½rs
s = rΘ



Example 1
A sector of a circle has arc length 10 cm and central angle 2.5 radians. Find its radius and area.
s = 11
Θ = 2.2
r = ??
You would use the formula to find the radius s = rΘ.
Plug into the formula and you get that r = 10/2.5= 4.
To find the area, you could use either K = ½r²Θ or K = ½rs .
K = ½(4²)2.5 = 20
or
K = ½(4)(10) = 20
Your answers would be r = 4 and K = 20



If a problem asks for apparent size, you should automatically assume to use the formula s = rΘ where;
r = distance between two objects
Θ = apparent size
s = diameter of an object.



Example 2
The Sun is 93,000,000 miles away from the Earth on average. The diameter of the Sun is about 865,000 miles in diameter. What is its apparent size?
Identify what's given.
s = 865,000
Θ = ??
r = 93,000,000
s = rΘ
Plug into the formula.
865,000 = (93,000,000) Θ
Θ = .009301075
The suns apparent size is .009301075 radians or like .5° if you are into degrees.