Angle Converter

Convert degrees, radians, gradians with visual and trigonometry

Angle conversion translates between degrees (360° in a full circle), radians (2π in a full circle), and gradians (400 grad in a full circle), with the fundamental relationship 180° = π radians ≈ 3.14159 radians.

Quick Presets

Convert Angle

Right Angle

Exactly 90°, forms an L shape

90° 0′ 0.0″

DMS (° ′ ″)

π/2

Fraction of π

0.00°

Complementary

90.00°

Supplementary

Trigonometric Functions

sin(θ)

1.0000

cos(θ)

0.0000

tan(θ)

∞

csc(θ)

1.0000

sec(θ)

∞

cot(θ)

0.0000

Common Angles

0°

30°

π/6

45°

π/4

60°

π/3

90°

π/2

120°

2π/3

135°

3π/4

150°

5π/6

180°

270°

3π/2

360°

2π

~57.3°

1 rad

Frequently Asked Questions

How do I convert degrees to radians?

Multiply degrees by π/180 (approximately 0.01745) to get radians. Example: 90° × π/180 = π/2 ≈ 1.5708 radians. Common values: 0° = 0 rad, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. Radians are the natural unit in mathematics and physics — they make calculus formulas for trigonometry simpler.

What is a radian and why is it used instead of degrees?

1 radian is the angle at the center of a circle that subtends an arc equal in length to the radius. Since circumference = 2πr, a full circle = 2π radians ≈ 6.2832 radians. Radians are preferred in mathematics and physics because they make formulas like arc length (s = rθ), angular velocity (ω = θ/t), and derivatives of trig functions simpler — sin(x) has derivative cos(x) only when x is in radians.

What is a gradian and where is it used?

A gradian (grad or gon) divides a full circle into 400 equal parts, so 90° = 100 grad and 180° = 200 grad. Gradians were designed so that 1% grade (slope) = 1 grad, making surveying calculations easier. They are used primarily in surveying and civil engineering in some European countries. Converting: degrees × 10/9 = gradians; radians × 200/π = gradians.

What are the trigonometric values for the most common angles?

Key values: sin(0°)=0, sin(30°)=0.5, sin(45°)=√2/2≈0.707, sin(60°)=√3/2≈0.866, sin(90°)=1. For cosine, these values appear in reverse order: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=0.5, cos(90°)=0. Tangent equals sine/cosine: tan(45°)=1, tan(60°)=√3≈1.732, and tan(90°) is undefined (infinity).

What is the difference between complementary and supplementary angles?

Complementary angles add up to 90°. Supplementary angles add up to 180°. Example: 35° and 55° are complementary (35+55=90); 110° and 70° are supplementary (110+70=180). In a right triangle, the two non-right angles are always complementary. On a straight line, adjacent angles are supplementary. These relationships are fundamental in geometry and trigonometry proofs.

Copied!