What each unit actually measures, the exact conversion formulas in both directions, a quick-reference table of common angles, and when to use which one.
Published August 22, 2026 · Reviewed by the Ihsabha Editorial Team
Every angle can be measured in more than one unit, and the two that show up constantly in math, science, and engineering are degrees and radians. They describe the exact same rotation using two entirely different scales, which is exactly why mixing them up causes so many calculator errors — an angle typed as "90" means something completely different depending on which mode is active. This guide covers what each unit actually represents, how to convert cleanly between them, and which one to reach for depending on the situation.
A degree divides a full circle into 360 equal parts, so one degree is 1/360 of a complete rotation. This system traces back to ancient Babylonian astronomy, where a base-60 number system made 360 (a multiple of 60) a convenient way to divide up the circle of the year and the sky. Degrees are intuitive for everyday use — a quarter turn is 90°, a half turn is 180°, a full turn is 360° — which is why they dominate everyday contexts like navigation, construction, sports, and basic geometry taught in school. If you want to go further, How to Calculate a Tip: The Complete Tip Percentage Guide explains this point in more depth. Rather than working this out by hand every time, the Scientific Calculator can handle the whole calculation for you.
A radian is defined directly from a circle's own geometry rather than an arbitrary division: one radian is the angle created when the arc length along the circle's edge equals the circle's radius. Since the full circumference of a circle is 2π times its radius, a complete rotation measures exactly 2π radians — meaning a radian is a "natural" unit that comes straight from the relationship between a circle's radius and its circumference, with no arbitrary constant like 360 involved at all.
Radians aren't just an alternative convention — they're the unit that makes calculus involving trigonometric functions work cleanly. The derivative of sin(x) is cos(x) only when x is measured in radians; in degrees, an extra conversion constant (π/180) would need to appear in every single derivative and integral involving a trig function, cluttering formulas that are otherwise elegant. Because of this, radians are the default angle unit in calculus, physics, engineering, and virtually every programming language's math library, even though degrees remain more intuitive for everyday human use.
To convert an angle from degrees to radians, multiply by π/180:
radians = degrees × (π/180). For example, converting 90° gives 90 × (π/180) = π/2 radians, which is approximately 1.5708 radians. This formula comes directly from the fact that 180° and π radians both represent a half rotation, so degrees and radians are proportional to each other with π/180 as the conversion factor.
To convert an angle from radians to degrees, multiply by 180/π — the reciprocal of the degrees-to-radians factor:
degrees = radians × (180/π). For example, converting π/3 radians gives (π/3) × (180/π) = 60°. Note that the π terms cancel out algebraically whenever the radian value is already expressed as a fraction of π, which is why many common angles convert to clean, simple degree values. For a deeper look, our guide on How to Calculate Percentage Off: Discount & Sale Price Formula covers this in more detail.
Not every radian value is a clean multiple of π — sensor readings, programming output, and real-world measurements are often given as plain decimals. Converting 2.5 radians to degrees: 2.5 × (180/π) ≈ 2.5 × 57.2958 ≈ 143.24°. The constant 180/π ≈ 57.2958 is worth remembering, since it's the number of degrees in exactly one radian, and multiplying any radian value by it gives the equivalent in degrees directly.
The degree is by far the older of the two units, with roots in ancient Babylonian astronomy roughly 4,000 years ago. Babylonian mathematicians used a base-60 (sexagesimal) number system, and 360 — being close to the number of days in a year and cleanly divisible by 60 — became the standard way to divide the circle of the zodiac and, by extension, any circle. The radian, by contrast, is a much more recent invention: the concept was formalized in the 1870s by physicist James Thomson (brother of Lord Kelvin), and the term "radian" itself was first published in 1873. Radians emerged directly out of the needs of calculus and physics, which by the 19th century required an angle unit that behaved cleanly inside derivatives and integrals — something degrees, with their arbitrary 360-part division, could never provide. For an instant, practical check, the Scientific Calculator is ready to go.
Radians earn their keep in geometry formulas involving circles, not just calculus. The arc length of a circle is given by s = rθ, where r is the radius and θ is the angle in radians — a clean, direct formula with no extra conversion constant. The same calculation in degrees requires an additional factor: s = r × θ × (π/180). Sector area follows the same pattern: A = ½r²θ in radians, versus A = (θ/360) × πr² in degrees. In both cases, radians remove an unnecessary constant from the formula because the unit itself is already built from the circle's own radius, rather than an arbitrary external scale.
A third angle unit, the gradian (also called gon), divides a full circle into 400 equal parts rather than 360 or 2π, making a right angle exactly 100 gradians. Gradians were introduced during the French Revolution as part of an attempt to metricate angle measurement alongside the meter and the kilogram, aligning a quarter-turn with a clean power of 10. They never achieved wide adoption outside of surveying and some European civil engineering contexts, but most scientific calculators still include a GRAD mode alongside DEG and RAD for this reason — worth knowing about mainly so an unfamiliar "G" indicator on a calculator display isn't mistaken for an error.
Angular velocity — how fast something rotates — is a common place radians appear in applied problems. If a wheel completes 3 full rotations in 2 seconds, its angular velocity in radians per second is calculated by converting rotations to radians first (each rotation is 2π radians), then dividing by time: (3 × 2π) ÷ 2 = 3π ≈ 9.42 radians per second. Working the same problem in degrees would require converting the final answer back with the 180/π factor to compare it against any formula written in radians, which is exactly the extra step radians are designed to eliminate in physics and engineering calculations. This naturally leads to a related question, answered in How to Calculate Percentage Error: Formula, Steps & Worked Examples.
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | ≈0.5236 |
| 45° | π/4 | ≈0.7854 |
| 60° | π/3 | ≈1.0472 |
| 90° | π/2 | ≈1.5708 |
| 180° | π | ≈3.1416 |
| 270° | 3π/2 | ≈4.7124 |
| 360° | 2π | ≈6.2832 |
Radians make a useful shortcut possible in physics and engineering: for small angles, sin(θ) ≈ θ and tan(θ) ≈ θ, as long as θ is measured in radians. This is called the small-angle approximation, and it works because for small angles the arc of a circle and a straight line connecting its endpoints become nearly indistinguishable in length. For example, at θ = 0.1 radians, sin(0.1) ≈ 0.09983, which is within 0.2% of 0.1 itself — close enough that the approximation is used to simplify pendulum motion, optics, and structural engineering formulas without materially affecting accuracy. This approximation only holds in radians; substituting a degree value directly into θ ≈ sin(θ) gives a meaningless result, since the formula depends on radians being the "natural" unit tied to the circle's own geometry.
Every scientific calculator has a mode setting, usually reached through a MODE or SHIFT+SETUP menu, that determines whether trigonometric functions interpret angle inputs as degrees or radians. The active mode is almost always displayed as a small "D" or "R" indicator somewhere on the screen. Since the same numeric input produces a different trigonometric result in each mode, it's worth building the habit of glancing at that indicator immediately before entering any sin, cos, or tan calculation — this single check prevents the most common trigonometry error on any calculator. Ihsabha's scientific calculator guide covers the rest of the calculator's functions in detail.
Degrees remain the standard unit anywhere a human needs to intuitively judge or communicate an angle: navigation and compass bearings, construction and carpentry, surveying, astronomy coordinates, sports (like a skateboarding "360"), and virtually all geometry taught before calculus. Their familiarity and clean divisibility — 360 splits evenly into 2, 3, 4, 5, 6, 8, 9, 10, 12, and many other useful numbers — make them the practical choice whenever the audience is a person rather than a formula. Try the Scientific Calculator to run these numbers with your own figures.
Radians take over anywhere calculus, physics, or programming is involved. Angular velocity in physics is naturally expressed in radians per second, derivatives and integrals of trigonometric functions require radians to stay algebraically clean, and nearly every programming language's built-in trig functions (Math.sin() in JavaScript, math.sin() in Python, and equivalents elsewhere) expect radian input by default. Engineers working with rotational mechanics, signal processing, and wave functions also default to radians, since the underlying physics equations are written in that unit from the start.
Converting 135° to radians: 135 × (π/180) simplifies to 3π/4, since both 135 and 180 share a common factor of 45 — giving an exact decimal value of approximately 2.356 radians. Converting 5π/6 radians to degrees: (5π/6) × (180/π) = (5 × 180)/6 = 900/6 = 150°, with the π terms canceling cleanly since the radian value was already a fraction of π. Converting a less tidy value like 200° to radians: 200 × (π/180) = 10π/9 ≈ 3.491 radians, a case where the exact fractional form (10π/9) is generally preferred over the decimal in further calculations, since it avoids compounding rounding error in any later step. It's also worth reading Order of Operations (PEMDAS/BODMAS): The Complete Guide if this situation applies to you.
Angles aren't limited to the 0°–360° (or 0–2π) range — negative angles measure rotation in the opposite (clockwise) direction, and angles beyond a full turn simply keep adding rotations. A −90° angle points the same direction as 270°, since both describe a quarter turn clockwise from the starting position; in radians, this is −π/2 and 3π/2 respectively. Angles that point in the same direction despite different numeric values are called coterminal angles, and converting between degrees and radians works identically for them as for any other angle — the same π/180 and 180/π factors apply regardless of whether the angle is positive, negative, or greater than a full rotation.
Skip the manual conversion. Ihsabha's Scientific Calculator includes a full degree/radian toggle, so trigonometric functions compute correctly in whichever unit your problem uses.
Multiply the number of degrees by π/180. For example, 90° × (π/180) = π/2 radians, which is approximately 1.5708 radians.
Multiply the number of radians by 180/π. For example, π/2 radians × (180/π) = 90°.
Radians are the unit in which calculus formulas involving trigonometric functions take their simplest form, since a radian is defined directly from the circle's own geometry (arc length divided by radius) rather than an arbitrary 360-part division. Most higher mathematics, physics, and programming languages default to radians for this reason.
The calculator will still return a number, but it will be the trigonometric value of the wrong angle. For example, sin(90) in degree mode correctly returns 1, but sin(90) in radian mode returns about 0.894, since 90 radians is a completely different angle than 90 degrees.
Degrees are always marked with the ° symbol directly after the number, as in 45°, and can be further broken into arcminutes (′) and arcseconds (″) for extra precision in fields like astronomy and surveying, where 1° = 60′ and 1′ = 60″. Radians, by contrast, are usually written with no symbol at all when the value is a multiple of π, such as π/2 or 3π, since the presence of π already signals the unit; a plain decimal radian value like 1.57 is sometimes labeled "rad" explicitly to avoid being mistaken for a unitless number or a degree value. When reading a textbook or a calculator's output, the absence of a ° symbol next to a number is usually — though not always — a signal that the value is already in radians.
Degrees and radians are two rulers measuring the same rotation, built for different jobs — degrees for human intuition and everyday communication, radians for the clean algebra that calculus and physics depend on. The conversion between them only requires two constants, π/180 and its reciprocal 180/π, and the discipline of checking a calculator's angle mode before every trigonometric calculation eliminates the single most common source of error in the whole topic.
Work it out instantly with Ihsabha's Percentage Calculator and Percentage Change Calculator. For related reading on Ihsabha's blog, see Significant Figures: Rules, Examples & How to Round Correctly, Reverse Percentage: How to Find the Original Number and Markup vs. Margin: How to Calculate Each Percentage Correctly.