幾何学傾き計算機

The Geometry Slope Calculator computes the slope of a straight line through two given points. Slope is one of the most fundamental concepts of coordinate geometry — it measures steepness, defines what "parallel" and "perpendicular" mean algebraically, and bridges geometry with algebra (the y = mx + b form) and calculus (the slope of a curve at a point IS the derivative). This tutorial covers what slope means, how the calculator computes it, the parallel/perpendicular tests, and edge cases like vertical and horizontal lines.

The slope formula

Given two points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the slope of the line through them is:

m = (y₂ − y₁) / (x₂ − x₁)

Often described as "rise over run" — the vertical change (rise) divided by the horizontal change (run). The result tells you how many units the line moves up (or down) for every 1 unit it moves to the right.

What different slopes look like

• Positive slope (m > 0): line rises from left to right. A slope of 1 means 1 unit up per 1 unit right (45° angle). A slope of 2 is steeper; 0.5 is shallower.
• Negative slope (m < 0): line falls from left to right.
• Zero slope (m = 0): horizontal line. y is constant — no rise as you move right.
• Undefined slope (x₁ = x₂): vertical line. The denominator x₂ − x₁ is 0 — division by zero — so slope cannot be expressed as a real number. Vertical lines have no slope, not "infinite slope".

Worked examples

Example 1 — positive slope: P₁ = (2, 3), P₂ = (5, 9). m = (9 − 3) / (5 − 2) = 6 / 3 = 2. Line rises 2 units for every 1 unit right.

Example 2 — negative slope: P₁ = (1, 4), P₂ = (3, 0). m = (0 − 4) / (3 − 1) = −4 / 2 = −2. Line falls 2 units per 1 unit right.

Example 3 — horizontal: P₁ = (0, 5), P₂ = (7, 5). m = (5 − 5) / (7 − 0) = 0 / 7 = 0. Flat line.

Example 4 — vertical (undefined): P₁ = (3, 0), P₂ = (3, 8). m = (8 − 0) / (3 − 3) = 8 / 0 = undefined. The line is vertical (x always equals 3).

Does the order of the points matter?

No — as long as you are consistent. If you swap which point is "1" and which is "2", both the numerator and denominator change sign. The two negatives cancel:

(y₁ − y₂) / (x₁ − x₂) = −(y₂ − y₁) / −(x₂ − x₁) = (y₂ − y₁) / (x₂ − x₁)

What you cannot do: subtract y₂ − y₁ in the numerator but x₁ − x₂ in the denominator. That would give a slope of the wrong sign.

Parallel lines

Two non-vertical lines are parallel if and only if they have the same slope:

m₁ = m₂

Geometric intuition: same rise/run means same direction of "lean". Two lines with equal slopes never intersect (unless they are the same line). Two vertical lines are also parallel — they share the trivial "undefined slope" classification.

Perpendicular lines

Two lines (neither vertical) are perpendicular if and only if their slopes multiply to −1:

m₁ × m₂ = −1

Equivalently: each slope is the negative reciprocal of the other. If m₁ = 2/3, then a perpendicular line has slope m₂ = −3/2.

Why this works: a line with slope m rotated 90° has slope −1/m. The negative comes from the orientation reversal (rise becomes run, run becomes rise); the reciprocal comes from swapping the roles of horizontal and vertical change.

Edge case: a horizontal line (m = 0) is perpendicular to a vertical line (undefined slope). The product 0 × (undefined) is also undefined, so the m₁ × m₂ = −1 rule is the limit of this case rather than a direct application.

From two points to the equation of the line

Once you have the slope, you can write the equation of the line in point-slope form:

y − y₁ = m(x − x₁)

Or rearrange to slope-intercept form: y = mx + b, where b = y-intercept.

Example: from P₁ = (2, 3) with slope m = 2: y − 3 = 2(x − 2), i.e., y = 2x − 1. The y-intercept is −1; at x = 0, y = −1.

Slope as the angle of inclination

If θ is the angle the line makes with the positive x-axis (measured counterclockwise), then:

m = tan(θ)

So a 45° line has slope tan(45°) = 1. A 60° line has slope tan(60°) = √3 ≈ 1.732. The connection between slope and angle is the foundation of every "angle of inclination" problem in physics (ramp angles, projectile launch angles, etc.).

Real-world applications

• Road and ramp design. A 10% grade highway has a slope of 0.10 (10 units rise per 100 units run). Wheelchair ramps follow ADA guidelines limiting slope to 1:12 (m ≈ 0.083).
• Stairs. Building codes specify maximum slope (rise/run) so stairs are walkable. A residential stair might be 7" rise / 11" run, giving m ≈ 0.636.
• Linear regression. Statistics fits a line through data points; the line's slope is the regression coefficient.
• Rate of change. In any time-series chart (price over time, temperature over time), slope is the rate of change. A stock chart's slope at a moment is the local price velocity.
• Calculus. The slope of a curve at a point IS the derivative at that point. Calculus extends "slope" from straight lines to arbitrary differentiable curves.

Common mistakes

• Subtracting in inconsistent orders. Use the SAME order for x and y. (y₂ − y₁) / (x₂ − x₁), not (y₂ − y₁) / (x₁ − x₂).
• Writing "infinite" for vertical-line slope. The slope is undefined, not infinite. There is no real number that satisfies the equation.
• Confusing the slope of x = 3 (vertical) with the slope of y = 3 (horizontal). Vertical has undefined slope; horizontal has slope 0.
• For perpendicular slopes, forgetting the negative. The product is −1, not +1. Negative reciprocal, not just reciprocal.