A radian is the angle that cuts off an arc as long as the radius.
Convert to radians
135°
Multiply by π180: 135 × π180 = 135π180 = 3π4, since 45 divides both. A quick check: 3π4 is three quarters of π, and π is 180°, so it is 135° ✓ D is 225° and A is 270° — both past the halfway mark, which 135° is not.
With a fixed start and a fixed direction, an angle is completely described by one number — which is what makes the whole of trigonometry possible.
−135° + 360° = 225°
add whole turns
225° is in quadrant III
between 180° and 270°
reference angle = 225 − 180 = 45°
the acute angle to the x-axis
The reference angle is always measured to the x-axis, never the y-axis, and it is always between 0° and 90°. It is what lets every trigonometric value be read off a first-quadrant fact plus a sign. In quadrant II subtract from 180°, in quadrant III subtract 180°, and in quadrant IV subtract from 360°: 340° has reference angle 360 − 340 = 20°.
s = rθ
θ in RADIANS
s = 12 × 2.5 = 30 cm
no π needed at all
A = ½r²θ = ½(144)(2.5)
the sector area
= 180 cm²
clean, because θ is in radians
Substituting 2.5° instead of 2.5 radians, or using 90 for a quarter turn, produces an arc length that is wrong by a factor of about 57. Convert before substituting, every time. These two formulas are the whole reason radians exist.
The same two conversion factors, in the three places they are most often skipped.
A radian measure with no π: convert 2 radians to degrees
2 × 180π = 360π
multiply by 180π
≈ 114.6°
π ≈ 3.1416
"2" with no degree sign means 2 radians. One radian is about 57.3°, so 2 radians is just under 115°.
An arc with the angle in degrees: 150° on a circle of radius 6 cm
θ = 150 × π180 = 5π6
convert FIRST
s = rθ = 6 × 5π6
now substitute
= 5π ≈ 15.7 cm
6 × 150 = 900 would be absurd
A question in degrees still needs radians inside s = rθ.
A large angle in radians: the angle between 0 and 2π coterminal with 17π4
17π4 − 2π = 17π4 − 8π4 = 9π4
one full turn off
9π4 − 8π4 = π4
a second turn off
In radians a full turn is 2π = 8π4 here; subtract it until the angle is below 2π.
| degrees | 30 | 45 | 60 | 90 | 180 | 270 | 360 |
|---|---|---|---|---|---|---|---|
| radians | π6 | π4 | π3 | π2 | π | 3π2 | 2π |
Read the denominator as "how many of these fit in a half turn". π6 is a sixth of 180°, so 30°. π4 is a quarter, so 45°. That reading converts most exam angles without any arithmetic at all.
Tests ask for a degree–radian conversion, a coterminal or reference angle, or an arc length or sector area. The fastest route: multiply by π180 or 180π, add or subtract whole turns of 360° or 2π, and convert to radians before using s = rθ or A = ½r²θ. A 10-second check: π6 is 30°, so compare your radian answer with that benchmark and see it is about the right size.
Find the arc length
A circle has radius 8 cm. What is the length of the arc cut off by an angle of π3 radians?
s = rθ = 8 × π3 = 8π3 cm, about 8.4 cm. C squares the radius, which belongs to the sector AREA formula, not the arc length. D converts π3 to 60 and multiplies, which is the degrees mistake this lesson warns about.
Hint: the angle is already in radians, so use it exactly as given. Arc length uses r once; sector area uses r².
