Quadrilateral Angle Calculator

The Quadrilateral Angle Calculator solves for the missing fourth angle of any four-sided figure when you know the other three. It is built on a single fact: the interior angles of any quadrilateral sum to 360°. This tutorial proves that fact, walks through how to find a missing angle from the other three, and explains how the same principle specializes to squares, rectangles, parallelograms, rhombuses, kites, and trapezoids.

Why the interior angles sum to 360°

Take any quadrilateral and draw one of its diagonals (a line segment connecting two opposite vertices). The diagonal cuts the quadrilateral into two triangles. The interior angles of each triangle sum to 180° — a basic theorem of plane geometry. Two triangles, each contributing 180°, total:

180° + 180° = 360°

The same proof works for any simple (non-self-intersecting) quadrilateral, regardless of whether it is convex or concave. As long as you can draw a single diagonal that stays inside the figure, the two-triangle decomposition works. For concave quadrilaterals you may need to pick the diagonal carefully, but the total stays 360°.

Solving for the missing angle

Given any three angles A, B, C of a quadrilateral, the fourth is:

D = 360° − (A + B + C)

The calculator handles this in either direction — enter the three you know and leave the unknown blank.

Worked examples

Example 1: A = 80°, B = 100°, C = 90°. D = 360° − (80 + 100 + 90) = 360° − 270° = 90°. A quadrilateral with three angles summing to 270° has its fourth angle exactly 90° — common in problems involving a right corner plus two known angles.

Example 2: A = 110°, B = 75°, C = 60°. D = 360° − 245° = 115°.

Example 3 — invalid input check: A = 200°, B = 100°, C = 100°. Sum already = 400° > 360°. The calculator returns an error because no real interior angle for D could make a valid quadrilateral. Either the input values are wrong or the figure has a reflex angle (greater than 180°) — see the concave section below.

Special quadrilaterals — the angle pattern simplifies

For a parallelogram see the dedicated Parallelogram Angle Solver.

Convex vs concave quadrilaterals

A convex quadrilateral has all four interior angles less than 180°. Both diagonals lie entirely inside the figure. The 360° interior sum applies in the most straightforward way.

A concave quadrilateral has one interior angle greater than 180° (called a reflex angle). Examples include arrowhead shapes and "dart" quadrilaterals. The interior sum is still 360° if you measure the reflex angle correctly — i.e., from inside the figure, taking the angle that exceeds a straight line.

Most middle-school problems assume convex quadrilaterals, so all four input angles are between 0° and 180°. If you have a concave figure and a reflex vertex, double-check that you are recording the interior (reflex) value, not the exterior (non-reflex complement).

Exterior angles

An exterior angle at a vertex is the supplement of the interior angle: exterior = 180° − interior. The four exterior angles of a convex quadrilateral always sum to 360° — this is a special case of the general "exterior angles of any simple polygon sum to 360°" theorem, which makes interior + exterior sums work out to (interior sum) + (exterior sum) = n × 180° where n is the number of sides.

Common mistakes

• Using 180° instead of 360°. 180° is the triangle interior sum, not the quadrilateral sum. A quadrilateral has twice as many vertices, so twice the angle sum.
• Mixing degrees and radians. Our calculator expects degrees. 360° = 2π radians; if your problem uses radians, convert first.
• Reading the reflex angle as 180° − reflex. If a problem says "the interior angle at this vertex is 220°", do not subtract 220 from 360 — 220° is the angle you should plug in directly. The interior of a concave vertex really is more than 180°.
• Forgetting which side is "consecutive" in a parallelogram. Consecutive = next to each other along the boundary. Opposite = diagonally across. Adjacent = next to. The angle relationships only hold for the correct pair.