A clock hand sweeping around is the perfect model for angles. As the hand moves, it sweeps an angle — measured in degrees or radians — and its tip traces an arc. The longer the hand, the longer the arc for the same angle. Drag the slider below to see it all live.
Angle = how far the hand has turnedArc = distance the tip travelsBigger radius = longer arc, same angle1 full turn = 360° = 2π rad
ℹ️ How to use: Drag the Angle slider to sweep the clock hand and watch the arc grow. Drag the Hand length slider to change the radius. Values update live.
🕐 The Clock — drag the slider →
⚙️ Adjust the Hand
0°
Degrees
0.000
Radians
0.0
Arc length (units)
0.00
Full turns
Arc swept0%
0Full circle = 2πr = 503 units
Key insight: Arc length = r × θ (in radians)
Change the hand length → arc changes, angle doesn't.
Same angle, bigger clock = longer arc. Same ratio: arc/r = θ
The clock connection: A clock hand completes 1 full turn (360° = 2π rad) in 12 hours (hour hand) or 60 minutes (minute hand). The minute hand covers 6° = π/30 rad per minute. At 3:00, the hour hand is at 90° = π/2 rad. At 6:00: 180° = π rad.
Clock angles — degrees
Time
Hour hand
Minute hand
12:00
0° = 0 rad
0° = 0 rad
3:00
90° = π/2
0° = 0 rad
6:00
180° = π
0° = 0 rad
9:00
270° = 3π/2
0° = 0 rad
12:30
15° = π/12
180° = π
Arc traced by clock hands
A typical clock has a minute hand of about 10 cm. In one hour:
θ = 2π rad (full turn)
Arc = r × θ = 10 × 2π = 62.8 cm per hour
In 15 min (π/2 rad): arc = 10 × π/2 = 15.7 cm
💡 This is why radians are natural: arc length = radius × angle (in radians). No extra conversion needed. In degrees you'd have arc = r × θ × π/180.
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What is a Radian?
A radian is defined by the arc of a circle. When the arc length equals the radius, the angle at the centre is exactly 1 radian. No arbitrary numbers — it comes straight from the geometry of circles.
Why use radians?
Degrees are arbitrary (360 was chosen by ancient astronomers). Radians are natural — they come directly from the circle itself.
arc = r × θ (θ in radians — no factor needed!)
In degrees: arc = r × θ × π/180 (awkward!)
💡 All engineering and physics formulas for rotation, waves, and circular motion use radians. Your calculator's radian mode is essential.
Key radian facts
1 radian ≈ 57.296°
360° = 2π ≈ 6.283 rad
180° = π ≈ 3.142 rad
90° = π/2 ≈ 1.571 rad
1° = π/180 ≈ 0.01745 rad
📌 π (pi) = 3.14159... It is irrational — never ends. In radian problems, leave answers in terms of π where possible.
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Converting Between Degrees and Radians
One formula, two directions. The key is π rad = 180° — use it as a multiplier.
Degrees → RadiansMultiply by π/180
radians = degrees × π/180
Convert 120° to radians
Formula: rad = 120 × π/180
Simplify: = 120π/180 = 2π/3
Answer:2π/3 ≈ 2.094 rad ✅
Convert 225° to radians
Formula: rad = 225 × π/180
Simplify: = 225π/180 = 5π/4
Answer:5π/4 ≈ 3.927 rad ✅
Radians → DegreesMultiply by 180/π
degrees = radians × 180/π
Convert 3π/4 to degrees
Formula: deg = (3π/4) × 180/π
Cancel π: = 3 × 180/4 = 540/4
Answer:135° ✅
Convert 2.5 rad to degrees
Formula: deg = 2.5 × 180/π
Calculate: = 450/π = 450/3.1416
Answer:143.2° ✅
⚠️ Always check your calculator is in the correct mode. For radian calculations use RAD mode. For degree calculations use DEG mode. Mixing them up is the most common error!
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Arc Length
The arc length is the distance along the curved edge of a sector. It is the most direct use of radians — the formula is beautifully simple.
Arc length formula: s = r × θ (θ must be in radians) or s = rθ
Step-by-step method
1Convert θ to radians if given in degrees: θ(rad) = θ° × π/180
2Apply s = rθ — multiply radius by the radian angle
3Check units — arc length has same units as radius
Three more examples
r=8m, θ=π/3 rad. Find s.
Already rad: θ=π/3
s=rθ: s = 8 × π/3 = 8π/3
Answer:s = 8.38 m ✅
s=15cm, r=10cm. Find θ.
Rearrange: θ = s/r = 15/10
θ in rad: 1.5 rad
In degrees:85.9° ✅
🥧
Sector Area
A sector is a "pie slice". Its area is a fraction of the full circle's area — and the formula uses radians naturally.
Sector area: A = ½r²θ (θ in radians) | Also: A = ½rs (where s = arc length)
Where the formula comes from
Full circle area = πr²
Sector is fraction θ/2π of full circle
A = πr² × θ/2π = r²θ/2 = ½r²θ ✓
💡 In degrees: A = (θ°/360°) × πr² — messier. In radians: A = ½r²θ — clean!
Summary of both formulas
Arc length: s = rθ
Sector area: A = ½r²θ = ½rs
Both need θ in RADIANS
📌 If θ is in degrees, convert first: multiply by π/180
Worked example 1
r=6cm, θ=2π/3. Find area and arc length.
Arc: s = 6 × 2π/3 = 4π = 12.57 cm
Area: A = ½ × 36 × 2π/3 = 12π
Answer:s=12.6cm, A=37.7cm² ✅
Worked example 2
A=50cm², r=8cm. Find θ and arc length.
Formula: 50 = ½ × 64 × θ
θ: θ = 100/64 = 1.5625 rad
In degrees: 89.5°
Arc:s = 8×1.5625 = 12.5 cm ✅
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Reference Table — Degrees to Radians
Memorise the key angles. All others can be derived from these.
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When to Use π and When to Use 180°
This confuses almost everyone. The rule is simple once you see it: π and 180° are the same angle — just written two different ways. Knowing when to use which form makes your working cleaner and faster.
Use 180° when…
✅ Converting: multiply by π/180 or 180/π
✅ Drawing or measuring real angles on paper
✅ SOHCAHTOA problems with given degree angles
✅ Bearing and navigation problems
✅ Angles in triangles (A+B+C = 180°)
✅ Clock and rotation described in everyday terms
✅ Calculator in DEG mode
180° = π rad = 3.14159... rad
Use π (radians) when…
✅ Arc length formula: s = rθ
✅ Sector area: A = ½r²θ
✅ Angular velocity: ω = 2πf or ω = 2πn/60
✅ Sine/cosine wave equation: y = sin(ωt)
✅ Period of a wave: T = 2π/ω
✅ Any formula from physics or engineering
✅ Calculator in RAD mode
π rad = 180° — exact, no decimal needed
The golden rule: If your formula has rθ, ½r²θ, ω, or any engineering symbol — use radians (π form). If you're describing an everyday angle, drawing a triangle, or working in degrees — use 180°.
When to write π vs 3.14159…
Leave answers in terms of π when the number is exact:
s = 6 × π/3 = 2π cm ← exact, leave as 2π
A = ½ × 25 × π/2 = 25π/4 cm² ← exact
Use 3.14159 (or press π on calculator) when you need a decimal answer:
s = 2π = 2 × 3.14159 = 6.28 cm
A = 25π/4 = 25 × 3.14159 / 4 = 19.63 cm²
💡 In exam: give exact answer first (in π), then decimal to 3 s.f. unless told otherwise.
Common mistakes to avoid
❌ Wrong: s = r × 180° (mixing degrees into rθ formula) ✅ Right: Convert first → s = r × π rad
❌ Wrong: Using 3.14 for π in exact answers ✅ Right: Write 2π not 6.28 until the final step
❌ Wrong: sin(90) in RAD mode expecting 1 ✅ Right: sin(90°) in DEG mode = 1, OR sin(π/2) in RAD mode = 1
💡 Memory trick: Degrees for drawing, Radians for calculating.
Side-by-side: same problem, two notations
Situation
Using 180° form
Using π form
A quarter turn
90°
π/2 rad
Arc, r=5, quarter turn
s = 5 × 90 × π/180 = 5π/2
s = 5 × π/2 = 5π/2 ✓ cleaner
sin of 60°
sin(60°) = 0.866 [DEG mode]
sin(π/3) = 0.866 [RAD mode]
Motor at 1500 RPM
25 rev/s × 360°/rev = 9000°/s
ω = 2π×25 = 157 rad/s ✓ used in v=rω
Full wave period
360° per cycle
T = 2π/ω ✓ engineering formula
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Using Your Calculator — DEG and RAD Modes
Your calculator has two trig modes. Using the wrong one gives completely wrong answers. Here is exactly how to check, switch, and use each mode on a Casio scientific calculator.
🧮 Casio fx-991MS — InteractiveDEG / RAD / SHIFT for inverses
ℹ️ SHIFT (turns amber) then sin/cos/tan gives sin⁻¹/cos⁻¹/tan⁻¹ — exactly like the real calculator. Orange labels above each key show what SHIFT does. Click D / R on the screen to switch DEG/RAD mode. Try sin(90) in DEG then switch to RAD to see why mode matters!
CASIO
fx-991MS 2nd edition
SOLAR
SHIFTM
0.
MODE: DEG
Try these — click to load & calculate
Check your mode FIRST, before every calculation. The display shows D for degrees or R for radians — usually at the top of the screen.
Casio — switching modes
On most Casio scientific calculators (fx-83, fx-85, fx-991):
📌 On Casio fx-83/85/991: π is usually SHIFT + ×10ˣ or a dedicated key. Check your model's manual.
💡 You can work in either mode for arc/sector calculations — just convert θ to radians manually first, then multiply. The trig functions (sin/cos/tan) are the only ones that care about the mode.
⚙️
Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
⚙️Mechanical:Angular velocity ω=θ/t. Belt contact arc s=rθ.
⚡Electrical:AC voltage V=V₀sin(ωt). ω=2πf rad/s.
🔧Manufacturing:CNC lathe angular feed rate in rad/s.
🌍Daily Life:Clock hand angle, pizza sector, satellite dish rotation.
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Engineering Applications
Radians and arc length appear everywhere in rotating machinery, robotics, civil engineering, and electronics.
⚙️ Rotational Speed — rad/s and RPM
Angular velocity ω is measured in rad/s. Converting from RPM (revolutions per minute):
At 60 km/h = 16.67 m/s: ω = v/r = 16.67/0.32 = 52.1 rad/s = 497 RPM
📡 Radar/Antenna — Sweep Arc
A radar dish sweeps through an angle. The width of the sweep zone at distance d is an arc — essential for coverage calculations.
Radar range r = 80 km Beam angle θ = 3° = 0.05236 rad
Arc width at range: s = 80 × 0.05236 = 4.19 km
For 360° sweep, total arc = circumference: s = 2π × 80 = 503 km
🌉 Bridge — Cable Sag as Arc
A suspension cable follows a catenary curve, approximated as an arc for small sag angles. The arc length gives the true cable length.
Span = 200m, sag angle θ = 0.15 rad Radius of curvature r = span/(2sinθ) r ≈ 200/(2×0.15) = 667 m
Arc (cable) length: s = r × θ = 667 × 0.15×2 = 200.1 m (slightly longer than span — as expected)
⚡ AC Circuits — Frequency to rad/s
Angular frequency ω links frequency f (Hz) to the sine wave equation. It is always in rad/s.
ω = 2πf rad/s
UK mains: f=50 Hz ω = 2π×50 = 314.2 rad/s
v = 325·sin(314.2t) volts
One cycle takes: T = 1/f = 0.02 s = 2π/ω rad of rotation → confirms ωT = 2π ✓
🏗️ CNC Machine — Rotary Table Angle
A CNC rotary table positions a workpiece. The controller uses radians internally but the operator enters degrees. Arc length gives the distance the cutting edge travels.
Rotary table r = 200mm Rotation: 72° = 72×π/180 = 2π/5 rad
Arc at rim: s = 200 × 2π/5 = 80π = 251.3mm
If cutting at this arc: feed_rate calculation uses arc (not chord) for accurate material removal Chord = 2r×sin(θ/2) = 400×sin(36°) = 235.1mm Arc is always longer than chord ✓
🌊 Wave Propagation — Phase and Wavelength
A wave travelling in space has a wavelength λ. The wave number k = 2π/λ (rad/m) connects distance to phase angle — radians are essential here.
Wave: y = A·sin(ωt − kx) ω = angular frequency (rad/s) k = wave number = 2π/λ (rad/m)
Sound in air: f=440Hz, v=343m/s λ = v/f = 343/440 = 0.780m k = 2π/0.780 = 8.06 rad/m
Phase shift over 0.5m: Δφ = k×x = 8.06×0.5 = 4.03 rad = 231°
🔩 Screw Thread — Lead and Helix Angle
A screw thread is a helix. One full rotation (2π rad) advances the screw by one lead distance. The helix angle uses arc length.
Pitch diameter d = 20mm → r = 10mm Lead (advance per turn) L = 3mm