Angle Guide: Degrees, Radians, Gradians & Tables

Everyday life speaks degrees, math speaks radians, soldiers and surveyors add mils and gradians. They all measure the same turn: 360° = 2π rad = 400 grad. Convert freely below, memorize the five angles that cover 90% of life, and browse the complete 0-360° ladder.

How to use the Angle Converter

1. Enter the angle.
2. Pick from- and to-units.
3. Check the key-angles table.

Formula

Key concepts, explained

Why radians rule math

Calculus cleans up: sin/cos derivatives and arc length (s = rθ) just work in radians. Degrees are human-sized (360 divides beautifully); radians are nature-sized. Convert at the boundary.

Mils, grads and turns

Artillery mils (6400/circle) make range math trivial (1 mil ≈ 1 m at 1 km). Gradians give right angles as 100 (surveying comfort). Turns (1 = full circle) simplify rotations in graphics and robotics.

Degrees inherit Babylonian 360

Three hundred sixty divides by 2, 3, 4, 5, 6, 8, 9, 10 and 12 - no other small number splits so evenly. Clocks (24 h), minutes and seconds descend from the same base-60 taste.

Reformers proposed 400 and 1000 divisions repeatedly; compatibility with every existing map, motor and protractor killed each attempt.

Radians measure arc directly

One radian subtends an arc exactly one radius long; the full circle runs 2π ≈ 6.2832. Arc length, sector area and angular velocity all collapse to multiplication.

Calculus demands radians: sine and cosine differentiate cleanly only in them, which is why physics abandons degrees at the classroom door.

Steradians go solid

A steradian carves a cone whose patch equals radius squared; the whole sphere spans 4π ≈ 12.57. Antenna gain, lumens and radiation all price per steradian.

One lumen equals one candela times one steradian - brightness follows the same geometry as angles.

Mils round true milliradians

A true milliradian circle holds 2π × 1000 ≈ 6,283 divisions; armies round to 6,400 for mental math at 1.9 % error. One mil still reads near one meter at one kilometer either way.

Binocular reticles and scope turrets speak mils: 0.1-mil clicks move impact 1 cm per 100 m.

Gradians carry the metric dream

Four hundred gradians per circle came from Revolutionary French metrication; right angles read an even 100. Surveyors in parts of Europe still stake in grads.

Every scientific calculator keeps a GRAD mode beside DEG and RAD - check it before trusting sines.

Minutes of latitude are nautical miles

One arc-minute of latitude spans 1,852 m by definition - the nautical mile itself. Seconds divide to about 30.9 m each.

A degree of latitude runs near 111 km everywhere; longitude degrees shrink toward the poles with the cosine.

Slope percent hides arctangents

Grade percent means rise over run times 100: 10 % converts via arctan(0.1) ≈ 5.71°. US interstates cap near 6 %; San Francisco streets hit 31.5 %.

Ski runs and roofs quote the same percent - always convert before comparing steepness claims.

Roof pitches decode simply

Twelve-twelve pitch means 12 rise over 12 run: arctan(1) = 45° exactly. Six-twelve gives 26.6°, four-twelve gives 18.4°.

Pitch names the rise with run fixed at twelve inches - carpenters frame by ratio, converters by degrees.

Clocks are angle machines

Minute hands sweep 6° per minute; hour hands creep 0.5° per minute (30° per hour). At 3:00 the gap stands exactly 90°.

Overlapping happens about eleven times per twelve hours - every 65.45 minutes, never on the hour except twelve.

Triangles package standard angles

The 30-60-90 triangle runs sides 1 : √3 : 2; the 45-45-90 runs 1 : 1 : √2. Memorize both and half of trig homework vanishes.

Drafting triangles embody them physically: 90-60-30 and 90-45-45 plastic, unchanged for a century.

Compass points slice evenly

Thirty-two named points divide the circle into 11.25° slices each: cardinal, intercardinal, then secondary-intercardinal quarters. Saying “north-northeast” once steered ships.

Modern compasses print every 2-5 degrees; sailors steer the lubber line, hikers the bezel.

Rotations prefer turns in code

Game engines and robots store full turns or radians and convert for display only. Angles wrap cleanly modulo one turn instead of exploding past 360.

Quaternions extend the idea to 3D, dodging gimbal lock that Euler degrees suffer.

Small angles linearize in radians

For tiny angles, sin θ ≈ θ and tan θ ≈ θ - but only in radians. Pendulum clocks and surveying approximations lean on this daily.

Error stays under a percent below about 14°; degrees need conversion first, always.

Sundials read hour angles

Earth turns 15° per hour, so sundial lines fan at 15° intervals (adjusted for latitude and longitude). Gnomons tilt parallel to the axis to stay accurate.

Equation-of-time corrections swing readings ±16 minutes through the year - sundials tell apparent, not civil, time.

Gears trade angle for torque

A 12-tooth gear driving 36 teeth cuts speed 3:1 - one input turn rotates output exactly 120°. Teeth ratios are angle ratios.

Bicycle gears do the same live: small chainring up front spins the cassette many turns per crank.

Protractors read both ways

Printed arcs carry inner and outer scales so either arm finds zero. Measure from the 0-line, not the plastic edge, and estimate tenths between marks.

Bevel protractors and digital finders read to minutes for machining and joinery.

Angular size rules vision

Sun and Moon both span about half a degree - the coincidence behind total eclipses. Thumbs at arm length cover near 2°; fists span about 10°.

Astronomers quote planet discs in arcseconds; Jupiter reaches about 50 at opposition.

Stairs code comfortable angles

A 7-inch riser with 11-inch tread climbs at arctan(7/11) ≈ 32.5° - inside the comfortable 30-37° band codes enforce. Steeper ships ladders switch to alternating treads.

Measure existing stairs before matching: one inch of riser changes the feel entirely.

Camera angles frame shots

A 50 mm lens on full frame covers about 40° diagonally - the “normal” view. Ultrawide zooms stretch past 100°; telephotos compress under 10°.

Focal length halves roughly double the angle only near wide ends - the math bends at extremes.

Slope stakes mark grades live

Graders read percent straight off rotating lasers: 2 % falls water away from slabs, 5 % maxes wheelchair ramps (1:12 with landings). Road crews paint targets, not degrees.

Laser plus rod beats string lines past ten meters - sag and wind corrupt strings silently.

Triangle angle sums never vary

Interior angles always total 180° on flat ground - measure two, know the third without climbing. Survey triangles close by this rule or re-shoot.

Spherical excess breaks it on planets: giant triangles exceed 180° by area over radius squared.

Machinists split degrees finely

Sine bars set angles from gauge blocks: sin θ = stack height ÷ 5-inch centers, reading to minutes. Bevel protractors verify to five minutes.

Compound angles need two setups in sequence - order matters because rotations do not commute.

Miter saw angles that close frames

Picture frames join 90-degree corners through 45-degree miters each side; eight-sided frames need 22.5-degree cuts that test blades divide precisely. Coping beats miters on inside baseboard corners where walls stray from square. Our converter bridges degree-radian settings on digital saws instantly.

Test cuts on scrap confirm angles before precious hardwoods commit. Glue plus pin nails hold miters while clamps dry. Cut long, sneak up.

Roof pitches spoken three ways

American carpenters say 6/12 where Europeans quote 26.6 degrees and engineers percent: all describe identical slopes our converter translates instantly. Steeper pitches shed snow while shallow ones resist wind better. Speed squares estimate materials where pitch multiplies areas.

Pitch gauges measure existing slopes directly for matching additions. Snow loads decide minimum pitches regionally. Speak all three dialects.

Wheel alignment cambers that wear tires

Camber tilts tops inward (negative) for cornering grip at straight-line wear cost; toe angles steer tire scrub dramatically per fraction of a degree. Caster trails steering stability invisibly. Alignment printouts quote degrees shops adjust on racks precisely.

Potholes knock settings out seasonally where annual checks pay in tire life. Our converter plays minor roles; geometry decides handling. Align yearly, rotate regularly.

Camera fields of view by focal length

Full-frame 24 mm spans 84 degrees diagonally where 50 mm narrows near 47 and 200 mm compresses near 12. Crop sensors multiply effective narrowness by 1.5 to 1.6 factors. Our converter bridges degree-radian lens math across calculators instantly.

Landscape wants wide, portraits flattering mid-tele, wildlife longest affordable. Test focal lengths by renting weekends before buying. Frame deliberately.

Drill point angles matched to metal

General-purpose bits grind 118 degrees where stainless wants 135 split points that walk less and cut cooler. Brad points spur wood cleanly at different geometries entirely. Our converter stays parked; protractors verify grinds approximately.

Drill doctors restore angles consistently where hand grinding varies. Pilot holes guide large bits straight. Match points to metal.

Knife bevels by task sensibly

Western chefs hold 20 degrees per side durably while Japanese blades run 15 keenly; axes split near 30 ruggedly. Guided systems hold angles repeatably where freehand varies with skill. Our converter bridges degree settings across guides instantly.

Stropping maintains edges between full sharpenings efficiently. Cutting boards preserve edges glass destroys. Match bevels to duties.

Stair risers, treads and code comfort

Comfortable stairs rise near 17 to 18 cm with 25 to 28 cm treads, totaling 45 in the classic 2R+T rule. Codes cap risers near 19 to 20 while nosings add usable depth. Steeper ship ladders suit lofts where space forbids comfort.

Stringers lay out through pitch blocks repeating identical steps precisely. Headroom minimums near 2 meters govern basement stairs strictly. Rise gently, tread generously.

Propeller pitch explained plainly

Pitch numbers state theoretical forward travel per revolution where slip reduces real progress substantially. Boats match pitch to engine RPM ranges for rated wide-open throttle. Our converter plays no direct role, but angular thinking mirrors length math conceptually.

Aluminum props forgive strikes where stainless performs efficiently. Test tachometers under load to prop correctly. Pitch for rated RPM.

Golf loft gaps that score

Four-degree gaps between wedges (48, 52, 56, 60) space full-shot distances evenly for most swings. Strong-lofted irons fly farther with lower trajectories demanding faster speeds. Fittings measure launch angles plus spin where static specs mislead.

Bent lies adjust direction more than lofts adjust distance. Gapping sessions on launch monitors beat range guessing. Gap wedges evenly.

Ski edge bevels tuned to snow

Side edges near 88 to 87 degrees (2 to 3 off 90) grip ice where 90 suits beginners forgivingly; base bevels near 1 degree release turns smoothly. Diamond stones maintain edges nightly on trips where files reset seasonally. Our converter bridges degree guides across tuning tools.

Detune tips and tails to prevent hookiness for intermediates. Wax to conditions after edges nightly. Tune edges often.

Compass bearings without confusion

Bearings count clockwise from north 0 to 360 where reciprocal back-azimuths add or subtract 180 for return legs. Magnetic declination offsets true north by degrees maps state explicitly. Our converter bridges mils, grads and degrees across military and hiking maps.

Triangulate with two-plus bearings crossing at positions. Practice in parks before wilderness needs. Trust compasses over hunches.

Sail trim angles that drive boats

Close-hauled sails trim near 10 to 15 degrees off centerlines while runs ease to 80 plus perpendicularly. Telltales stream both sides when angles balance perfectly. Apparent wind shifts demand constant trimming gusts dictate.

Travelers plus sheets share load where vangs control leech tension downwind. Ease in lulls, trim in puffs. Trim telltales streaming.

Solar panel tilt by latitude simply

Year-round fixed tilts near latitude degrees balance seasons where winter-plus-15 favors heating loads and summer-minus-15 favors cooling. Adjustable racks change seasonally where roofs allow access safely. Our converter plays minor roles; sun-path geometry decides.

Azimuth true south (north in southern hemisphere) matters more than exact tilt generally. Shade avoidance beats tilt perfection absolutely. Tilt near latitude, face equator.

Satellite dishes aimed patiently

Elevation plus azimuth plus skew angles from setup screens point dishes where meters beep strongest; fractions of degrees decide signal lock. Mast plumb precedes all adjustments absolutely. Our converter bridges degree inputs across receiver brands.

Trees grow into signal paths years after installs work perfectly. Tighten bolts progressively while peaking signals. Plumb masts, peak slowly.

Torque angles after snug sequences

Torque-to-yield bolts tighten to snug torque plus 90-degree turns that stretch fasteners into elastic zones precisely. Head gaskets sequence spirals outward in stages manuals specify exactly. Angle gauges slip over ratchets cheaply for accuracy.

Reusing TTY bolts risks failures where replacements cost little. Calibrate torque wrenches yearly for critical work. Snug first, angle second.

Chair ergonomics reclined slightly

Seat pans tilt near 5 degrees back with backrests near 100 to 110 from horizontal for spinal comfort across workdays. Monitor tops sit at eye level with arms supported at desk height neutrally. Our converter plays no role; protractors set showroom chairs initially.

Sit-stand desks alternate postures hourly where static perfection fails eventually. Lumbar rolls fill gaps stock chairs leave. Recline slightly, move hourly.

Telescope eyepieces and true fields honestly

Divide apparent field by magnification for true sky spans: 50-degree Plossls at 100x show half-degree moon-sized views precisely. Wide 82-degree eyepieces frame extended nebulae dramatically at equal powers. Our converter bridges degree-radian math behind optics instantly.

Exit pupils near 2 to 3 mm suit middle-aged eyes while youngsters use 7 fully. Dark skies matter more than eyepiece upgrades substantially. Match eyepieces to targets deliberately.

Binocular numbers decoded before buying

8x42 means 8 times magnification through 42 mm objectives where twilight factors and exit pupils decide low-light performance honestly. Higher magnification narrows fields while amplifying handshake miserably without support. Our converter plays minor roles; try before buying absolutely.

Roof prisms pack slim while Porro prisms depth-perceive cheaply. Waterproofing plus warranties decide field longevity. Glass quality beats magnification always.

Ladder angles that stay put

Extension ladders set near 75 degrees (1:4 lean ratio) grip reliably where steeper tips backward and shallower slides outward. Tie tops plus spiked feet on soft ground prevent both failures redundantly. Our converter turns ratio rules into degree checks instantly.

Three points of contact climb safely always; overreaching sideways topples secured ladders anyway. Level feet with leg levelers on slopes. Lean 1:4, tie top.

Wheelchair ramps at 1:12 that work

Accessible ramps rise 1 unit per 12 run (4.8 degrees) with landings every 9 meters where steeper DIY planks strand users dangerously. Slip-resistant surfaces plus edge curbs matter as much as slopes. Permits inspect commercial ramps strictly.

Portable suitcase ramps bridge single steps for visitors temporarily. Handrails both sides help broadly beyond wheelchairs. Build gentle, land level.

Sundial gnomons angled to latitude

Gnomons tilt at local latitude pointing true north so shadows sweep hour lines evenly through seasons. Horizontal dials suit gardens where vertical walls need declining designs calculated specifically. Our converter turns latitude degrees into build angles directly.

Longitude corrections plus equation-of-time tables refine readings to clock time precisely. Position dials level and oriented truly. Tilt to latitude, orient true.

Ceiling fan blade pitch that moves air

Blades pitch near 12 to 15 degrees balancing airflow against motor load where flat paddles stir nothing and steep ones stall noisily. Downrod lengths place fans 2.4 to 2.7 meters above floors for safety plus breeze. Our converter plays minor roles; pitch gauges verify after cleaning bends.

Reverse winter modes destratify ceiling heat usefully at low speeds. Balance kits calm wobbles weights cause. Pitch moderately, hang correctly.

Door hinge bevels that close cleanly

Hinge leaves mortise flush where screws seat fully without proud heads binding jambs seasonally. Bevel door edges 3 degrees on lock sides to clear frames gracefully. Our converter plays no role; sharp chisels decide fit.

Shim hinges with cardboard slivers to adjust reveals evenly. Pre-drill hardwoods to prevent splits absolutely. Mortise flush, bevel slightly.

Dovetail slopes that hold drawers

Softwoods dovetail near 1:6 slopes while hardwoods suit 1:8 finer angles resisting pull-out beautifully. Marking gauges plus sharp saws cut baselines cleanly where routers jig repetitive work. Our converter turns ratios into degrees for bevel gauges instantly.

Half-blind fronts hide end grain elegantly on drawer boxes. Practice poplar before walnut preciousness. Slope soft coarse, hard fine.

Chisel bevels plus micro-bevels that pare

Bench chisels grind 25 degrees primary with 30 micro-bevels honed quickly thereafter; paring chisels run lower near 20 delicately. Hollow grinds from wheels ease honing where flat stones then polish. Our converter bridges guide settings across systems.

Strop regularly between full sharpenings to maintain hair-popping edges. Mallet work wants steeper durable edges. Grind once, hone often.

Plane blades cambered subtly

Smoothing planes hone straight across with whisper camber preventing track marks; jack planes camber aggressively for thick shavings. Chip breakers set hair-close tame tearout dramatically. Our converter plays minor roles; sharpening guides set angles repeatably.

Sole flatness matters as much as blade sharpness for fine work. Wax soles glide measurably easier. Camber jacks, straighten smoothers.

Axe grinds by task sensibly

Splitting axes run 30-plus degree convex cheeks that pop wood apart while carving hatchets take 20-degree keen edges slicingly. Files restore profiles in fields where stones polish at home. Sheaths cover edges absolutely between uses.

Handle grain aligns with heads for strength where replacements need fitting patiently. Chop with legs clear, glance zones empty. Grind splitting blunt, carving keen.

Ski wax iron temps that glide

Wax irons run near 120 to 140 C by wax type where smoke means too hot degrading fluoros and bases. Keep irons moving steadily tip-to-tail without pausing over edges. Thermometers built into irons drift; wax melt behavior tells truth.

Scrape warm after cooling thoroughly, brush structures into bases progressively. Choose waxes by snow temperature, not air prestige. Iron moving, scrape cool.

Step-by-step walkthrough

Walkthrough: arc length

1. Radius 5 m, angle 60° → radians: π/3 ≈ 1.047.
2. Arc: 5 × 1.047 ≈ 5.24 m.
3. Sector area: ½ × 25 × 1.047 ≈ 13.1 m².

Walkthrough: 10% slope in degrees

1. Ratio: 10 ÷ 100 = 0.1.
2. Arctan: arctan(0.1) ≈ 5.71°.
3. Highway-plausible, driveway-normal.

Walkthrough: 6/12 roof pitch

1. Ratio: 6 ÷ 12 = 0.5.
2. Arctan: arctan(0.5) ≈ 26.57°.
3. Round: 26.6° - walkable with care.

Walkthrough: clock gap at 3:15

1. Minute hand: 15 × 6 = 90°.
2. Hour hand: 3 × 30 + 15 × 0.5 = 97.5°.
3. Gap: 97.5 − 90 = 7.5°.

Walkthrough: NYC latitude to decimal

1. Split 40°42′46″: 40 + 42/60 + 46/3600.
2. Compute: 40 + 0.7 + 0.0128.
3. Result: 40.7128° N.

Walkthrough: mils to meters at 2 km

1. Rule: 1 mil ≈ 1 m per km (6400 circle).
2. Multiply: 5 × 2 = 10.
3. Target spans about 10 m.

Walkthrough: gear output angle

1. Ratio: 36 ÷ 12 = 3:1 reduction.
2. One input turn: 360 ÷ 3.
3. Output rotates exactly 120°.

Worked examples

Complete reference tables

Degrees → radians → gradians

Landmarks: 90° is π/2, 180° is π, 360° is 2π.

Angle conversion matrix

1 deg equals 3,600 arcsec at the far end of the scale.

deg → rad (1-180)

deg → grad (1-180)

deg → rad (1-180)

rad → deg (1-50)

Key angles

Common mistakes to avoid

• Calculator in wrong mode (the eternal trig error).
• Mixing diameter and radius in arc math.
• DMS seconds > 60 (carry!).
• Assuming small-angle ≈ in degrees (radians only!).
• Degrees inside radian formulas - arc length needs radians first.
• Bearings measured counterclockwise like math class (compass runs clockwise from north).
• Reading 12-pitch roofs as 12 degrees (it is 45°).
• Averaging angles across 0°/360° arithmetically (359° and 1° average 180°, not 0° - use vectors).
• Mixing 6400 mils with true milliradians (1.9 % gap bites past 500 m).

Pro tips

• Check mode with sin(30): 0.5 = degrees.
• Sketch angles; orientation errors show visually.
• Navigation bearings: clockwise from north (unlike math!).
• Learn π/6, π/4, π/3, π/2 cold.
• Check radian mode with sin(π/6) = 0.5 instead.
• Memorize sine ladder 0/30/45/60/90: 0, 1/2, √2/2, √3/2, 1.
• Clocks overlap every 65.45 minutes - predict, then verify.
• Grades to degrees: 10% ≈ 6°, 20% ≈ 11°, 45% ≈ 24°.
• Calibrate phone compasses with figure-eight waves before trusting.

Where you will use this

Construction

Slopes, miters and roof pitches.

Navigation

Bearings, coordinates, great circles.

Graphics and robotics

Rotations love turns and radians.

Surveying

Gradians, chains and DMS coordinates stake land.

Astronomy

Arcseconds split planets; steradians meter light.

Aviation

Headings, runways numbered by tens of degrees.

Machining

Sine bars and bevel gauges work in minutes.

Glossary

Frequently asked questions

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