TRIGONOMETRIC & ANGULAR CONVERTER

Angle Calculator & Unit Converter

Q: What is the difference between complementary and supplementary angles?

Complementary angles are two angles that sum to exactly 90 degrees (forming a perpendicular right angle). Supplementary angles are two angles that sum to exactly 180 degrees (forming a continuous straight line).

Q: How do I convert degrees to radians manually?

Multiply the angle in degrees by π and divide by 180 (Radians = Degrees × π / 180). For example, 90° × π / 180 = π/2 ≈ 1.5708 radians.

Q: Why is the tangent of 90 degrees undefined?

In trigonometry, tangent represents the ratio of the opposite side to the adjacent side (or sin/cos). At 90 degrees, the adjacent side (cosine) equals zero. Dividing by zero is mathematically undefined, meaning the tangent approaches infinity.

Q: What is a reflex angle?

A reflex angle is an angle measuring strictly greater than 180 degrees and strictly less than 360 degrees. It represents the larger exterior rotation between two rays sharing a vertex.

An angle calculator converts angular measurements between degrees, radians, gradians, and turns while computing trigonometric ratios and angle pairs. Enter any angle in degrees or radians to instantly calculate complementary, supplementary, and reflex angles alongside quadrant coordinates.

Angle Calculator & Unit ConverterReady

DEG (°)RAD (rad)

Angle Value

DEG (°)

Input Unit

Presets:

Unit Conversions

Degrees:45.0000°

DMS:45° 0' 0"

Radians:0.7854 rad

Gradians:50.0000 gon

Turns:0.1250

Angle Relationships

Classification:Acute

Complement (90-θ):45.0000°

Supplement (180-θ):135.0000°

Reflex (360-θ):315.0000°

Primary Trig Functions

sin(θ):0.7071

cos(θ):0.7071

tan(θ):1.0000

Reciprocal Trig Functions

csc(θ) = 1/sin:1.4142

sec(θ) = 1/cos:1.4142

cot(θ) = 1/tan:1.0000

LOCAL COMPUTATION ONLYRESOLUTION: 0.01°

NO SERVER UPLOADS

How to Use This Tool

Select your input unit: Degrees (°), Radians (rad), Gradians (gon), or Turns/Revolutions.
Enter your numerical angle value into the input field, or tap one of the quick benchmark buttons.
Inspect the live output card for instant conversions to all other angular units and DMS (Degrees, Minutes, Seconds).
Review the calculated angle relationships (Complementary, Supplementary, Reflex) and trigonometric functions (sin, cos, tan, cot, sec, csc).
Use the 'Copy Results' button to paste all calculated values directly into your project notes.

Estimate only. Verify with professional tools for critical work.

What the Angle Calculator Does

The Angle Calculator & Converter is a multi-system trigonometric utility designed to translate angular measurements between four fundamental coordinate systems: Sexagesimal Degrees (°), SI Radians (rad), Centesimal Gradians (gon), and Fractional Turns. In addition to unit conversion, the tool evaluates geometric angle relationships (complementary, supplementary, reflex) and calculates all six basic trigonometric ratios (sin, cos, tan, csc, sec, cot).

Engineers, draftsmen, surveyors, and students routinely encounter angles expressed in varying formats—such as nautical degrees-minutes-seconds (45° 30′ 00″), calculus radians (0.7854 rad), or surveying grads (50 gon). This instrument harmonizes those representations without requiring separate calculators or manual conversion steps.

Formulas for Angular Conversion & Trigonometric Ratios

Angles represent rotational displacement around a central vertex. The mathematical relationships between angular units originate from the ratio of circular circumference to radius:

Degrees → Radians

rad = θ° × (π / 180°)

Multiplies degree value by π radians per half revolution (180°).

Radians → Degrees

θ° = rad × (180° / π)

Multiplies radian value by the conversion factor ≈ 57.2957795°.

Degrees → Gradians

gon = θ° × (200 / 180) = θ° / 0.9

Gradians divide a right angle into 100 metric decimal grads.

Degrees → Turns

Turns = θ° / 360°

Represents fraction of a complete 360° circular rotation.

Two Worked Examples

Example 1: Converting a 60° Angle to Radians, Gradians, and Angle Pairs

Input: θ = 60.0°.

Radians: rad = 60 × (π / 180) = π / 3 ≈ 1.0472 rad.
Gradians: gon = 60 / 0.9 = 66.6667 gon.
Complementary Angle: 90° - 60° = 30.0°.
Supplementary Angle: 180° - 60° = 120.0°.
Reflex Angle: 360° - 60° = 300.0°.
Trig Values: sin(60°) = √3/2 ≈ 0.8660, cos(60°) = 0.5000, tan(60°) = √3 ≈ 1.7321.

Example 2: Resolving a Nautical Survey Angle (2.3562 Radians)

Input: An autonomous underwater glider logs an angular yaw deflection of 2.3562 radians (3π/4).

Convert to Degrees: θ° = 2.3562 × (180 / π) = 135.000°.
Classification: Because 90° < 135° < 180°, this is an obtuse angle in the second quadrant.
Complementary Angle: 90° - 135° = -45° (Geometric complement does not exist for angles > 90°).
Supplementary Angle: 180° - 135° = 45.0°.
Trig Values: sin(135°) = √2/2 ≈ 0.7071, cos(135°) = -√2/2 ≈ -0.7071, tan(135°) = -1.0000.

How to Find Reference Angles and Coterminal Angles

In trigonometry, calculus, and circular motion, angles on the Cartesian coordinate plane are often analyzed using reference angles and coterminal rotations:

Reference Angle (θ_ref)

A reference angle is the positive acute angle (0° ≤ θ_ref ≤ 90°) formed between the angle's terminal side and the horizontal x-axis:

Quadrant I (0°–90°): θ_ref = θ
Quadrant II (90°–180°): θ_ref = 180° - θ
Quadrant III (180°–270°): θ_ref = θ - 180°
Quadrant IV (270°–360°): θ_ref = 360° - θ

Coterminal Angles

Coterminal angles share the exact same initial side (positive x-axis) and terminal ray, differing only by full 360° revolutions:

θ_coterminal = θ ± (360° × k), where k is an integer

For example, angles coterminal with 45° include 405° (45° + 360°) and -315° (45° - 360°).

Standard Benchmark Angles Reference Table

The following reference table summarizes conversions and trigonometric functions across fundamental benchmark angles:

Degrees	Radians	Complement	Supplement	Reflex	sin(θ)	cos(θ)	tan(θ)
0°	0.0000	90°	180°	—	0.0000	1.0000	0.0000
30°	0.5236	60°	150°	330°	0.5000	0.8660	0.5774
45°	0.7854	45°	135°	315°	0.7071	0.7071	1.0000
60°	1.0472	30°	120°	300°	0.8660	0.5000	1.7321
90°	1.5708	0°	90°	270°	1.0000	0.0000	Undefined
120°	2.0944	—	60°	240°	0.8660	-0.5000	-1.7321
135°	2.3562	—	45°	225°	0.7071	-0.7071	-1.0000
150°	2.6180	—	30°	210°	0.5000	-0.8660	-0.5774
180°	3.1416	—	0°	180°	0.0000	-1.0000	-0.0000
270°	4.7124	—	—	90°	-1.0000	-0.0000	Undefined
360°	6.2832	—	—	—	-0.0000	1.0000	-0.0000

Common Mistakes When Calculating Angles

Mismatched Calculator Modes: Entering degree values into scientific software set to Radian mode (or vice versa). Evaluating sin(90) in radian mode yields 0.8940 instead of 1.0000. Always check your active angle mode.
Division by Zero at Asymptotes: Tangent is undefined at odd multiples of 90° (90°, 270°). Cosecant is undefined at 0° and 180°. Computer programs will crash or return Infinity / NaN if unchecked.
Confusing Complementary with Supplementary: Remember that complementary angles sum to 90°, while supplementary angles sum to 180°. Angles greater than 90° cannot have a positive real complement.
Overlooking Coordinate Signs: In Quadrant II (90° < θ < 180°), cosine and tangent are negative while sine is positive. Ensure vector directions reflect proper quadrant signs.

REFERENCE & SUPPORT

Frequently Asked Questions: Angle Calculator

Clear answers to common questions about angle calculations, protractors, accuracy, and geometry.

What is the difference between complementary and supplementary angles?

Complementary angles are two angles that sum to exactly 90 degrees (forming a perpendicular right angle). Supplementary angles are two angles that sum to exactly 180 degrees (forming a continuous straight line).

How do I convert degrees to radians manually?

Multiply the angle in degrees by π and divide by 180 (Radians = Degrees × π / 180). For example, 90° × π / 180 = π/2 ≈ 1.5708 radians.

Why is the tangent of 90 degrees undefined?

In trigonometry, tangent represents the ratio of the opposite side to the adjacent side (or sin/cos). At 90 degrees, the adjacent side (cosine) equals zero. Dividing by zero is mathematically undefined, meaning the tangent approaches infinity.

What is a reflex angle?

A reflex angle is an angle measuring strictly greater than 180 degrees and strictly less than 360 degrees. It represents the larger exterior rotation between two rays sharing a vertex.

How We Calculate

Angular conversions are computed using exact trigonometric ratios (rad = θ × π / 180) and standard unit circle definitions. Trigonometric values handle domain asymptotes safely, returning undefined where division by zero occurs.

Last updated: March 2026 • Verified against standard trigonometric identities.

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