How to Calculate Angles Between Two Lines

By Kevin Lee
When lines cross, angles form.

When two non-parallel lines cross, they create an angle between them. If the lines are perpendicular, they form a 90-degree angle. Otherwise, they create an acute, obtuse or other type of angle. Every angle has a "slope." For instance, a ladder against a wall has a slope whose value varies according to the angle of the ladder. Using a little geometry, you can compute the angle between two intersecting lines by determining their slopes.

Compute Slopes

Draw two non-parallel lines on a sheet of graph paper. Label the lines "Line A" and "Line B."

Draw a small circle at any point on "Line A." Note its x and y coordinates on the graph paper and call the coordinates x1 and y1. Assume that x1 is 1 and y1 is 2.

Draw another small circle at another location on the line. Note the coordinates, and call them x2 and y2. Assume that x2 is 3 and y2 is 4.

Write down the following slope equation.

Slope_A = (y2-y1)/(x2-x1)

Plugging in the sample values for the coordinates, you get this equation:

Slope_A = (4-2)/(3-1)

The value for Slope_A is 1 in this example.

Repeat these steps and compute the slope of "Line B." Label that slope "Slope_B." For this example, assume that the value for "Slope_B" is 2.

Compute Angle

Write down the following equation:

Tangent_of_Angle = (SlopeB - SlopeA) / (1+SlopeA*SlopeB)

Perform the calculation. The equation looks as follows using the values computed in the previous section:

Tangent_of_Angle = (2-1) / (1+1*2)

In this example, the value for "Tangent_of_Angle" is 0.33.

Use the trigonometry table to find the angle whose tangent is "Tangent_of_Angle" as computed previously. If you look up the example value, 0.33, you discover that its corresponding angle, to the nearest 10th of a degree, is 18 degrees. The angle between "Line A" and "Line B" is 18 degrees.

Tip

If you do not have a trigonometry table, you can find one online.

About the Author

After majoring in physics, Kevin Lee began writing professionally in 1989 when, as a software developer, he also created technical articles for the Johnson Space Center. Today this urban Texas cowboy continues to crank out high-quality software as well as non-technical articles covering a multitude of diverse topics ranging from gaming to current affairs.