How this instrument works
Any simple quadrilateral — a square, a kite, an arrowhead-shaped concave figure, or a four-sided lot surveyed with no particular symmetry — has interior angles that add up to exactly 360°. That is the quadrilateral angle-sum theorem, and it is the four-sided sibling of the far more famous fact that a triangle's three angles always total 180°: fix any three of the four corners and the last one is not a measurement you take, it is a number the other three have already decided.
The proof needs only one extra line, drawn once. Pick any vertex and connect it to the opposite corner it is not already joined to; that single diagonal slices the quadrilateral into exactly two triangles, each worth 180° on its own, and the pair adds straight to 360°. It is the smallest case of a wider pattern — cut any simple n-sided polygon this way and you always get (n − 2) triangles — but a quadrilateral is the first shape on that ladder that actually needs a cut: a triangle is already whole with zero diagonals drawn.
The fourth angle is pinned down the moment three are chosen, but the shape itself is not: a rectangle and a squashed parallelogram can share all four angle readings while looking nothing alike, because angles alone say nothing about side length. The theorem still holds for concave quadrilaterals, provided the one reflex corner — greater than 180° — is entered as such rather than as its smaller supplementary look-alike; miss that distinction and the fourth angle comes out wrong even though the arithmetic itself is fine.
- Enter the first interior angle you already know into the Angle 1 field, in degrees by default (radians and turns are also selectable).
- Enter the second known angle into Angle 2, and the third into Angle 3 — order does not matter, only that all three share one unit.
- Read the fourth interior angle straight off Angle 4 (computed); it updates the instant all three inputs are filled in.
- If one corner is reflex, greater than 180°, enter it as that full reflex measurement rather than its smaller supplement, so the 360° total stays correct.
Worked example — 80°, 90°, and 100° known
A four-sided plot of land is surveyed with three corners measured at 80°, 90°, and 100° — 1.3962634015954636, 1.5707963267948966, and 1.7453292519943295 radians. Angle 4 (computed) returns 360° − 80° − 90° − 100° = 90°, or 1.5707963267948966 radians, closing the plot into a valid simple quadrilateral with no gap and no overlap at the fourth corner.
That fourth corner could not have come out any other way: 80 + 90 + 100 + 90 = 360 exactly, the fixed total every simple quadrilateral's angles must reach. Swap the third reading from 100° to a steeper 120° and Angle 4 would drop to 70° in response — the sheet always hands back whatever the first three angles still owe the full 360°.
Questions
What is the interior angle sum theorem for quadrilaterals?
Every simple quadrilateral's four interior angles add up to exactly 360°, whether it is a square, a kite, or an irregular four-sided shape. Split it with one diagonal into two triangles, each worth 180°, and the two totals combine to the fixed 360° — the reason this sheet only needs three angles to find the fourth.
How is this different from the triangle's 180° rule?
It is the same idea one side later. A triangle's three angles sum to 180° because it is already a single triangle; a quadrilateral needs one diagonal to become two triangles, doubling the total to 360°. A fifth side, a pentagon, needs two diagonals for three triangles and 540°, following the same (n − 2) × 180° pattern for any n-sided polygon.
Can I use this for a concave quadrilateral, like an arrowhead shape?
Yes, as long as the reflex angle — the one where the shape caves inward — is entered as its true measurement above 180°, not the smaller angle it might resemble at a glance. Enter it correctly and the four angles still total 360°; enter the smaller look-alike instead and the fourth angle comes out wrong by exactly the amount you shorted it.
If I know all four angles, does that also fix the side lengths?
No — angles alone never pin down a quadrilateral's size or side lengths. A small square and a much larger rectangle can share all four 90° angles while differing completely in area and perimeter; this sheet only ever answers the angle question, so pair it with a separate side-length or diagonal calculation for the rest.
Why enter only three angles instead of four?
Because the fourth is never free information — once three interior angles of a simple quadrilateral are fixed, the 360° total decides the last one automatically. Typing in a fourth value yourself would either just repeat this sheet's answer or, if it disagreed, describe a shape that cannot actually close.
Does the 360° rule still hold for a quadrilateral drawn on a globe?
No — 360° is a flat-plane result. A four-sided figure drawn on a curved surface such as a sphere has interior angles that sum to more than 360°, growing with the enclosed area. This calculator assumes an ordinary flat quadrilateral, which covers maps, floor plans, and nearly every practical surveying case.