Complementary Angles Calculator

Background.

Two angles are complementary when they add up to a right angle, 90°. Knowing one is enough to know the other, and that small fact turns up more often than its simplicity suggests.

The most common place is a right triangle. Because a triangle's angles total 180° and the right angle takes 90 of them, the two acute angles must share the remaining 90 — if one is 35°, the other is 55°, with no measuring needed. The same split appears whenever something leans against a vertical: a ladder at 75° to the ground sits at 15° to the wall, and the sun 30° above the horizon is 60° from the point directly overhead, the angle astronomers call the zenith angle. Geographers call 90° minus a latitude the colatitude.

Trigonometry is built on the pairing. The cosine is, by name and by definition, the sine of the complement: cos θ = sin(90° − θ). Tangent and cotangent, secant and cosecant pair up the same way, which is why a table of sines from 0° to 90° doubles as a table of cosines read from the bottom up.

The page works in decimal degrees and accepts any known angle from 0° to 90°, returning 90° minus it. An angle larger than 90° has no complement among positive angles; what you probably want then is its supplement, the angle that completes 180°.

What is complementary angles calculator?

Complementary angles are a pair of angles whose measures add to exactly 90 degrees, or π/2 radians. The complement of an angle θ is 90° − θ, so 20° and 70° are complementary, and so are 45° and 45°. The two angles need not touch or share a vertex; when they are adjacent, sharing a side, together they form a right angle. Only an angle between 0° and 90° has a positive complement.

How to use this calculator.

1. 01Enter the angle you know in decimal degrees, between 0 and 90. If it is in radians, multiply by 180/π first: 0.6 rad is about 34.38°.
2. 02If the angle comes in degrees and minutes, divide the minutes by 60 so that 35°20′ becomes 35.333…°, or subtract by hand from 89°60′, which gives 54°40′ directly.
3. 03Read the other angle. In a right triangle it is the remaining acute angle; for a ladder, ramp or roof it is the same slope measured from the vertical instead of the horizontal.

The formula.

Other angle = 90° − known angle. The 90 is the size of a right angle, a quarter turn; it is the definition of complementary rather than a measured constant, so the result is exact for whatever you enter. The relationship is a straight line with slope −1: every degree added to one angle comes off the other, so the pair always move in opposite directions and meet at 45°, the only angle that is its own complement. Entering 0° returns 90° and entering 90° returns 0°. Those end points are degenerate cases, since textbooks usually define complementary angles as two positive angles, but the subtraction still holds. In radians the same rule reads π/2 − θ.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.