One conversion factor, and two formulas that refuse to work without it.
Convert to radians
Write 150° in radians.
Multiply by π180: 150 × π180 = 5π6 after cancelling by 30. B is the fraction upside down. A quick sanity check: 150° is just under a half turn, and 5π6 is just under π.
Lay the radius along the rim. The angle it opens up is one radian.
circumference = 2πr
the whole rim
2πr ÷ r = 2π
how many radii fit around
2π rad = 360°
a full turn, two ways of saying it
So radians are not a new kind of number — they are a length divided by a length. That is why they carry no unit symbol.
π rad = 180°
× π180
the π goes on top
× 180π
the π goes underneath
Both are just π180 = 1, written the way that cancels. Going to radians, the degrees must cancel — so 180 sits underneath. Going back, the π must cancel, so π sits underneath.
150 × π180
degrees to radians
150π180
before simplifying
= 5π6
divide both by 30
And in reverse:
7π6 × 180π
radians to degrees
= 7 × 30
π cancels, 180 ÷ 6 = 30
= 210°
third quadrant
5π6 is the answer. Turning it into 2.618 loses the exact form the choices are written in, and costs you time you do not have.
| Degrees | Radians | Where it lands |
|---|---|---|
| 30° | π6 | first quadrant |
| 45° | π4 | first quadrant |
| 60° | π3 | first quadrant |
| 90° | π2 | quarter turn |
| 120° | 2π3 | second quadrant |
| 180° | π | half turn |
| 270° | 3π2 | three quarters |
| 360° | 2π | full turn |
Read the denominator as a slice of 180. Sixths are 30° each, quarters are 45° each — so 5π6 is five thirty-degree steps, 150°.
s = rθ
arc length, θ in radians
s = 12 × 5π6
12 ÷ 6 = 2
s = 10π
≈ 31.4 units
A = ½r²θ
sector area
A = ½ × 144 × 5π6
r² = 144
A = 72 × 5π6 = 60π
72 ÷ 6 = 12, times 5
s = 12 × 150 = 1800 is the classic wreck. Both formulas are built on the radian definition, so the angle must be converted before it goes in.
Convert 2.5 radians to degrees.
2.5 × 180π = 450π
nothing cancels this time
≈ 143.2°
1 rad ≈ 57.3°
Different: with no π in the radian measure, the π stays in the denominator. 2.5 rad is not 2.5 × 180 = 450°.
Convert −225° to radians.
−225 × π180 = −225π180
the sign just rides along
= −5π4
÷ 45 top and bottom
Different: a negative angle turns clockwise. Convert the size as usual and keep the minus sign.
Convert 22.5° to radians.
22.5π180 = 45π360
× 2 top and bottom clears the decimal
= π8
half of π4
Different: a decimal degree. Clear the decimal before simplifying — or notice 22.5° is half of 45°.
A sector has arc length 4π and area 20π. Find its central angle in radians. The bridge is A = ½rs — area is half the radius times the arc.
A = ½r²θ = ½r(rθ) = ½rs
since s = rθ
20π = ½ × r × 4π
put in what we know
20π = 2πr, r = 10
÷ 2π
θ = s ÷ r = 4π ÷ 10 = 2π5
72°
Check: ½ × 100 × 2π5 = 20π ✓
20π ÷ 4π = 5 is r2, not θ. Find r first, then θ = sr.
Stories use the same two formulas. A clock hand that sweeps 15 minutes turns a quarter turn, π2; a wheel turning 540° turns 3π. Use the radius, not the diameter, in s = rθ.
Expect a conversion inside a story (clock hands, wheels), an arc or sector with the angle in degrees that must be converted first, or a sector given backwards. Keep π in the answer and use s = rθ, A = ½r²θ or A = ½rs. Ten-second check: the sector’s fraction θ2π of the full circle should give the same arc and area.
150° is 150360 of a full turn — that fraction of the rim, and of the area.
Find the central angle
A sector of radius 9 has arc length 6π. Find its central angle in radians.
Arc length is s = rθ when θ is in radians, so 6π = 9θ and θ = 6π9 = 2π3. A multiplies instead of dividing. The formula is this clean only in radians — that is the whole reason radians exist.
