Angle from Three Points Calculator

How do you find an angle from three points?

Three points define an angle: the middle point is the vertex and the other two show the arms. In this calculator the vertex is point B and the result is angle ABC.

First write the vectors from the vertex to the other two points: BA = A − B and BC = C − B. Then use the dot product: cos β = (BA · BC) / (|BA| × |BC|). The dot product is the sum of the products of matching components.

Example

For A(3, 0), B(0, 0) and C(3, 3), BA = (3, 0) and BC = (3, 3). The dot product is 3 × 3 + 0 × 3 = 9 and the lengths are 3 and 4.243. So cos β = 9 / (3 × 4.243) = 0.7071 and β = 45 degrees.

Other angles and the area of the triangle

The same steps at vertices A and C give the other two angles, and the three add up to 180 degrees. The area of the triangle is |x₁·y₂ − y₁·x₂| / 2 in the plane and half the magnitude of the cross product in space.

Angles in space

With three dimensional points the formula stays the same, with the z component added to the dot product and the lengths. The angle between two pipes, two steel members or two cables meeting at a node is found this way.

Where is it used?

Deflection angles in a traverse, checking from coordinates whether a corner is square, reading an elbow or bend angle off a drawing, and joint angles of a robot arm all rest on this calculation.

Limits

The result lies between 0 and 180 degrees. The angle is undefined when the vertex coincides with one of the other points. With very short arms, a small measuring error in the coordinates changes the angle noticeably.