Complementary Angles Calculator

What Are Complementary Angles?

In Euclidean geometry, two angles are called complementary angles if the sum of their angle measures equals exactly $90^\circ$ ($\frac{\pi}{2}$ radians, or $100$ gradians). When two complementary angles share a common vertex and a common side, they form a right angle corner.

Complementary angles play an indispensable role in geometry, trigonometry, engineering, carpentry, and physics. This online Complementary Angles Calculator allows you to calculate the missing complement of any angle or test whether any two given angles are complementary.

The Complementary Angle Formula

If two angles are denoted as $\alpha$ and $\beta$, their relationship is given by:

$$\alpha + \beta = 90^\circ$$

To find the unknown complementary angle $\beta$ when $\alpha$ is known:

$$\beta = 90^\circ - \alpha$$

Expressed across different angular measurement units:

Degrees ($^\circ$): $\beta = 90^\circ - \alpha^\circ$
Radians (rad): $\beta = \frac{\pi}{2} - \alpha \approx 1.5708 - \alpha$
Gradians (grad): $\beta = 100 - \alpha$

Adjacent vs. Non-Adjacent Complementary Angles

Complementary angles do not necessarily have to be next to each other:

Adjacent Complementary Angles: Share a common vertex and ray, forming an exact $90^\circ$ right angle (such as the perpendicular corner of a wall or a square).
Non-Adjacent Complementary Angles: Are separated in space but still add up to $90^\circ$. A classic example is the two acute angles inside any right-angled triangle.

Trigonometric Relationships of Complementary Angles

Complementary angles are directly tied to cofunction trigonometric identities:

$$\sin(\alpha) = \cos(90^\circ - \alpha) = \cos(\beta)$$ $$\cos(\alpha) = \sin(90^\circ - \alpha) = \sin(\beta)$$ $$\tan(\alpha) = \cot(90^\circ - \alpha) = \cot(\beta)$$

For dedicated trigonometry conversions, visit our Cofunction Calculator and Angle Converter.

Complementary Angles vs. Supplementary Angles

Property	Complementary Angles	Supplementary Angles
Total Sum	$$90^\circ \ (\pi/2 \text{ rad})$$	$$180^\circ \ (\pi \text{ rad})$$
Geometric Shape Formed	Right angle (perpendicular corner)	Straight line ($180^\circ$ linear pair)
Individual Angle Type	Must both be acute ($< 90^\circ$)	Can be two right angles or one acute + one obtuse
Example Pair	$$35^\circ \text{ and } 55^\circ$$	$$110^\circ \text{ and } 70^\circ$$

Frequently Asked Questions

Can two obtuse angles be complementary?

No. An obtuse angle is strictly greater than $90^\circ$. Since complementary angles must sum to exactly $90^\circ$, neither angle can exceed $90^\circ$, meaning two complementary angles in standard geometry must both be acute ($0^\circ < \theta < 90^\circ$).

Can a right angle have a complementary angle?

The complement of a $90^\circ$ right angle is $0^\circ$ ($90^\circ - 90^\circ = 0^\circ$).

How do you remember the difference between complementary and supplementary angles?

A helpful mnemonic is alphabetical and numeric order: C comes before S in the alphabet, just as 90° (Complementary) comes before 180° (Supplementary). Another memory trick is "C is for Corner ($90^\circ$)" and "S is for Straight ($180^\circ$)".

What is the complement of 45 degrees?

The complement of $45^\circ$ is $90^\circ - 45^\circ = 45^\circ$. An angle of $45^\circ$ is equal to its own complement.

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