What Are Complementary Angles?
Complementary angles are two angles whose measures add up to exactly 90° (π/2 radians). In other words, if you have angle α, its complementary angle β satisfies:
α + β = 90° = π/2 rad
β = 90° − α
The word complementary comes from Latin complementum, meaning "that which completes." Complementary angles are most commonly encountered in right triangles, where the two non-right angles are always complementary.
How to Find a Complementary Angle
Finding a complementary angle is straightforward:
- In degrees: Subtract the angle from 90°.
Complementary angle = 90° − α - In radians: Subtract the angle from π/2.
Complementary angle = π/2 − α
Example: If α = 30°, then the complementary angle = 90° − 30° = 60°.
In radians: π/2 − π/6 = π/3 ≈ 1.0472 rad.
Non-Adjacent and Adjacent Complementary Angles
Complementary angles do not need to be adjacent (next to each other). They only need to sum to 90°. However, when they are adjacent, they form a perfect right angle together.
- Adjacent complementary angles share a common vertex and side, forming a 90° right angle.
- Non-adjacent complementary angles are separate angles that simply sum to 90°.
Complementary vs. Supplementary Angles
| Property | Complementary | Supplementary |
|---|---|---|
| Sum equals | 90° (π/2 rad) | 180° (π rad) |
| Example | 30° and 60° | 110° and 70° |
| In a right triangle | Yes — the two acute angles | No |
| On a straight line | No | Yes — two angles on a straight line |
Measurement Systems
This calculator supports two measurement systems:
- Degrees (°) — US/Imperial system: The most common system in everyday use. A full rotation is 360°, and a right angle is 90°. Complementary angles sum to 90°.
- Radians (rad) — Metric/SI system: The standard unit in mathematics and physics. A full rotation is 2π rad, and a right angle is π/2 ≈ 1.5708 rad. Complementary angles sum to π/2 rad.
Conversion: 1 radian = 180°/π ≈ 57.2958° | 1° = π/180 ≈ 0.017453 rad
Complementary Angles in Right Triangles
In any right triangle, the two non-right angles are always complementary. If one acute angle is α, the other is (90° − α). This relationship gives rise to the co-function identities in trigonometry:
- sin(α) = cos(90° − α)
- cos(α) = sin(90° − α)
- tan(α) = cot(90° − α)
- sec(α) = csc(90° − α)
Common Complementary Angle Pairs
| Angle α (degrees) | Angle α (radians) | Complement (degrees) | Complement (radians) |
|---|---|---|---|
| 0° | 0 | 90° | π/2 ≈ 1.5708 |
| 15° | π/12 ≈ 0.2618 | 75° | 5π/12 ≈ 1.3090 |
| 20° | π/9 ≈ 0.3491 | 70° | 7π/18 ≈ 1.2217 |
| 30° | π/6 ≈ 0.5236 | 60° | π/3 ≈ 1.0472 |
| 45° | π/4 ≈ 0.7854 | 45° | π/4 ≈ 0.7854 |
| 60° | π/3 ≈ 1.0472 | 30° | π/6 ≈ 0.5236 |
| 75° | 5π/12 ≈ 1.3090 | 15° | π/12 ≈ 0.2618 |
| 90° | π/2 ≈ 1.5708 | 0° | 0 |
Frequently Asked Questions
- Can an angle be complementary to itself?
- Yes — 45° is the only angle that is self-complementary (45° + 45° = 90°).
- Can obtuse angles be complementary?
- No. Since complementary angles must sum to 90°, both angles must be acute (between 0° and 90°).
- Can two right angles be complementary?
- No. Two right angles sum to 180°, not 90°.
- What is the complement of 0°?
- The complement of 0° is 90°. And conversely, the complement of 90° is 0°.