Degrees and radians are two rulers for the same thing, the size of an angle, and the calculator can work in either one. The unit decides what the number means. Convert with × π/180 one way and × 180/π the other, and a flip of the DEG/RAD toggle changes how every trig input you type is read.
Key Takeaways
30°in radians is30 × π / 180 = 0.523598775598, andπ/4radians back to degrees is45.- A radian is the angle whose arc equals one radius, so a full turn is
2πradians rather than360.- The calculator defaults to DEG, where
sin(30)→= 0.5; in RAD mode the same expression is the sine of 30 radians, about-0.988031624093.- In RAD mode,
sin(30*pi/180)→= 0.5, because the conversion put the angle in radians explicitly.
What is a degree, and what is a radian?
Both measure angle, and neither one is mathematically forced. A degree is a historical choice: one degree is 1/360 of a full turn. A radian is geometric. Take the radius of a circle, lay it along the circumference, and the angle it sweeps at the centre is one radian. Because a full circumference is 2πr, a full turn is exactly 2π radians, and half a turn is π radians. That last fact, π radians equals 180°, is the only conversion you need to remember. The pi explained post looks at what the constant itself is.
Why radians exist
Radians exist because they keep the geometry clean. Along a circle of radius r, the arc length is simply s = r × θ when θ is in radians. In degrees the same formula drags a factor of π/180 along. That same factor would follow through every derivative and series expansion. The derivative of sine is cosine only when the angle is in radians; in degrees it picks up an extra constant. The degrees vs radians guide goes deeper into that, and the trigonometry reference collects the identities and values that assume one unit or the other.
Converting degrees to radians
Multiply by π/180. For 30° that is 30 × π/180, and 30pi/180 → = 0.523598775598. The answer comes out as a plain number; radians stay attached only by label. 60° gives 60 × π/180 = π/3, and 60pi/180 → = 1.0471975512. If you keep the expression on one line, the calculator does the conversion for you.
Converting radians to degrees
Multiply by 180/π. For π/4 that is (π/4) × 180/π = 45, and (pi/4)*180/pi → = 45. The π cancels top and bottom, which is why the familiar angles come out as whole numbers. π/2 radians is 90°, and (pi/2)*180/pi → = 90. The conversion turns any angle in radians into the degree reading on the same circle.
Common angles in both units
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 30° | 0.523598775598 |
| 45° | 0.785398163397 |
| 60° | 1.0471975512 |
| 90° | 1.57079632679 |
Each radian value is degrees × π/180, shown to the calculator’s 12 significant digits. π itself displays as 3.14159265359, so every number in the table is a scaled piece of it. The sines and cosines of these angles sit in the trigonometry reference, which is the usual place to look them up.
What the DEG/RAD toggle actually changes
The toggle does not change what an angle is. It changes how the trig functions read the number you type. In DEG mode, sin(30) → = 0.5, because 30 is read as 30 degrees. In RAD mode, the same expression returns the sine of 30 radians, about -0.988031624093. The calculator function reference lists every function that honours the toggle, and the calculator remembers your choice between visits.
When you want an answer that does not depend on the toggle, write the conversion into the expression. In RAD mode, sin(30*pi/180) → = 0.5, because the arithmetic turned the angle into radians itself. In DEG mode the same line would treat that small number as degrees, so check which mode you are in before you trust it. The degrees vs radians guide shows the same examples from the page side.
Every trig function on the keypad, sine, cosine, tangent, and their inverses, reads the same toggle, so the choice affects whole expressions at once. The practical guidance is simple: keep DEG for school geometry, construction, and most everyday problems, and use RAD for calculus and anything that feeds a series or a derivative. The angle mode is remembered between visits, so an answer that worked last week still behaves the way you left it.
Frequently asked questions
What is the quickest way to check which mode the calculator is in?
Evaluate sin(30). In degrees the answer is 0.5. In radians it is about -0.988031624093. This is the standard way to tell the modes apart.
How do I convert 60 degrees to radians?
Multiply by π/180: 60 × π/180 = π/3, and 60*pi/180 → = 1.0471975512.
Why do mathematicians prefer radians?
Because the arc length formula and the derivatives of the trig functions come out clean. sin θ differentiates to cos θ only when θ is in radians; in degrees you drag a constant factor around.
Do 30 degrees and 0.5236 radians name different angles?
No. They name the same angle on two rulers. 30° and about 0.5236 radians are the same wedge of a circle, just measured in different units.