Enter slopes of two lines to calculate the intersection angle
Line L\u2081 (Slope k\u2081)
k\u2081 =
Or line equation Ax+By+C=0
Line L\u2082 (Slope k\u2082)
k\u2082 =
k\u2082 \u2260 k\u2081
Examples: 0.5, -1/3 \u2248 -0.333, \u221a2 \u2248 1.414. Just enter the slope value.
Result
Angle between L\u2081 (k\u2081=) and L\u2082 (k\u2082=)
Step-by-Step Derivation
Angle Formula
tan \u03b8 = |(k\u2082 \u2212 k\u2081) / (1 + k\u2081\u00b7k\u2082)|
\u03b8\u2208[0\u00b0, 90\u00b0]. When k\u2081\u00b7k\u2082 = \u22121, tan \u03b8 \u2192 \u221e, meaning the lines are perpendicular (90\u00b0). When k\u2082 = k\u2081, the angle is 0\u00b0 (parallel).
⚠When k\u2081\u00b7k\u2082 = \u22121 (i.e., k\u2082 = \u22121/k\u2081), the two lines are perpendicular, angle is always 90\u00b0. No need to compute the denominator.
Properties of the Angle Between Two Lines
The angle between two lines is the acute (or right) angle formed at their intersection. It reflects the difference in the lines' slopes.
Parallel
When k\u2081 = k\u2082, lines are parallel (or coincident), angle = 0\u00b0. Denominator 1+k\u2081k\u2082 \u2260 0, but numerator k\u2082\u2212k\u2081 = 0, so tan \u03b8 = 0.
Perpendicular
When k\u2081\u00b7k\u2082 = \u22121, lines are perpendicular, angle = 90\u00b0. Denominator 1+k\u2081k\u2082 = 0, tan \u03b8 \u2192 \u221e, arctan(\u221e) = 90\u00b0.
Acute/Obtuse
With absolute value, tan \u03b8 \u2265 0, \u03b8 \u2208 [0\u00b0, 90\u00b0]. Without absolute value, arctan ranges in (\u221290\u00b0, 90\u00b0), distinguishing relative orientation.
Relation to Slope
The greater the slope difference, the larger the angle. When k\u2081=0 (horizontal), angle = |arctan(k\u2082)|. Opposite signed slopes tend to produce larger angles.
Why take absolute value? What is the angle range?▼
Absolute value gives the acute (or right) angle. \u03b8\u2208[0\u00b0,90\u00b0], where 0\u00b0=parallel, 90\u00b0=perpendicular. For directional orientation (0\u00b0~180\u00b0), use the formula without absolute value.
How to calculate with general form Ax+By+C=0?▼
Slope k = \u2212A/B (B\u22600). Substitute into formula. If B=0 (vertical), slope is infinite: k\u2081\u2192\u221e, tan \u03b8 = |1/k\u2082| (k\u2082\u22600) or \u03b8=90\u00b0 (k\u2082=0).
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