Slope Calculator
Find the slope between two points, write line equations from slope and a point, and identify parallel and perpendicular lines. Shows rise/run, angle, and distance.
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How It Works
Slope = (y2−y1)/(x2−x1). Rise is the vertical change, run is the horizontal change. Positive slope goes up right; negative goes down right.
m = (y2−y1)/(x2−x1)(1,2) and (4,8): m=(8−2)/(4−1)=6/3=2Given slope m and one point (x1,y1): b = y1−m·x1. Then write y = mx + b. This converts point-slope to slope-intercept.
b = y1 − m·x1, then y = mx + bm=2, point (1,2): b=0, y=2xParallel lines have equal slopes. Given a line with slope m, any line y=mx+c (different c) is parallel. They never intersect.
Parallel: same m, different by=3x+2 parallel to y=3x−5Perpendicular slopes multiply to −1: m2 = −1/m1. If a line has slope 4, perpendicular slope = −1/4.
m_perp = −1/mm=3 → m_perp=−1/3Convert slope to angle: θ = arctan(m). A slope of 1 gives 45°, slope 0 gives 0°, steep lines approach 90°.
θ = arctan(m) × 180/πm=1 → θ=45°; m=2 → θ≈63.4°d = √((x2−x1)²+(y2−y1)²). The Pythagorean theorem applied to horizontal and vertical components.
d = √(Δx²+Δy²)(0,0) to (3,4): d=√(9+16)=5Quick Reference
Common examples — verify instantly above.
Basic
(0,0) to (4,8)
m = 2
Negative
(1,5) to (3,1)
m = −2
Horizontal
(2,3) to (7,3)
m = 0
Vertical
(4,1) to (4,5)
undefined
Parallel
m=3 parallel
m=3
Perp.
m=3 perpendicular
m=−1/3
Angle
m=1
45°
Distance
(0,0)→(3,4)
5
Tips & Shortcuts
Remember: slope = rise/run. Count squares on a grid — up 3, right 2 = slope 3/2.
Perpendicular slopes: flip and negate. Slope 4/3 → perpendicular is −3/4.
A horizontal line always has slope 0. A vertical line always has undefined slope.
To check if a point lies on a line: substitute its coordinates into y=mx+b. If both sides are equal, the point is on the line.
The slope of a line through the origin (0,0) and (a,b) is simply b/a.
Steeper lines have larger absolute slope values. A slope of 10 is much steeper than a slope of 2.
Common Mistakes
Computing (x2−x1)/(y2−y1) instead of (y2−y1)/(x2−x1)
Slope is always rise (y change) over run (x change), never the reverse.
Thinking a zero slope means undefined
Zero slope is a horizontal line — perfectly valid. Undefined slope is a vertical line (division by zero).
Using the wrong point for y-intercept calculation
Use the full formula b = y1 − m·x1 with the actual coordinates of any point on the line.
Assuming parallel means same line
Parallel lines have equal slopes but different y-intercepts. The same slope AND same intercept means identical lines.
Getting perpendicular slope wrong for fractions
For slope 3/4, perpendicular is −4/3 (flip AND negate). Just flipping to 4/3 is wrong.
Forgetting to simplify the slope fraction
Slope 6/4 should be simplified to 3/2. Always divide by GCF.
Frequently Asked Questions
Slope measures steepness: rise over run = (y2−y1)/(x2−x1). Positive slope goes up left-to-right; negative goes down.
Undefined slope occurs on vertical lines where x1=x2. Division by zero makes the slope undefined.
Parallel lines have identical slopes. Two lines are parallel if and only if their slopes are equal and they have different y-intercepts.
Perpendicular lines have slopes that are negative reciprocals: m1 × m2 = −1. If slope is 3, perpendicular slope is −1/3.
y = mx + b, where m is slope and b is y-intercept. It is the most common way to write a line equation.
For y=mx+b, slope is m directly. For ax+by=c, rearrange to y=(−a/b)x+c/b, so slope = −a/b.
Step 1: slope m = (y₂−y₁)/(x₂−x₁). Step 2: use point-slope form y − y₁ = m(x − x₁). Step 3: rearrange to slope-intercept y = mx + b. Example: points (2,3) and (6,7). m = (7−3)/(6−2) = 1. y − 3 = 1(x − 2) → y = x + 1. Verify with (6,7): 7 = 6+1 ✓. Standard form: x − y + 1 = 0 (or Ax + By + C = 0 with integer coefficients).
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