Line through (2, 3) and (5, 11)
m = (11 - 3) / (5 - 2) = 8/3 ≈ 2.67. b = 3 - (8/3) × 2 = 3 - 16/3 = -7/3 ≈ -2.33. Equation: y = (8/3)x - 7/3. Angle: arctan(8/3) ≈ 69.4°. Steeply rising line.
Slope Calculator - Бесплатный онлайн-калькулятор
Calculate the slope (gradient) of a line through two points, plus its y-intercept, equation in slope-intercept form and angle with the x-axis. Used in algebra, calculus, surveying and engineering.
- Формула на странице
- Работает в браузере
- Ввод остаётся приватным
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Slope (m)
2
Δy / Δx = 12 / 6
Δx (run)
6
Δy (rise)
12
Angle with x-axis
63.434949°
Y-intercept (b)
-1
y = mx + b
Line equation
y = 2.0000x − 1.0000
Часть содержимого пока показывается на английском.
Что такое Slope Calculator
Slope (or gradient) is one of the most useful single concepts in mathematics - it measures how steeply something rises or falls. In coordinate geometry it's the slope of a line through two points; in calculus it generalises to the derivative of a curve at a point; in real life it appears as road grade percentage (highway signs say '8% gradient ahead'), roof pitch ratios for monsoon drainage, staircase rise/run for code-compliant design, ramp inclines for wheelchair accessibility (max 1:12 per accessibility standards), and stock chart trend lines.
Class 9-10 mathematics introduces slope through coordinate geometry; Class 11-12 builds it through trigonometric angles and derivatives; JEE/NEET physics uses it for motion graphs (slope of position-time graph = velocity, slope of velocity-time graph = acceleration). This calculator computes slope from two points along with the line's y-intercept, complete equation in slope-intercept form (y = mx + c), angle with the positive x-axis, and special-case handling for vertical and horizontal lines.
Как пользоваться: Slope Calculator, шагов: 3
Enter the coordinates of point 1.
Enter the coordinates of point 2.
Read slope, intercept, equation and angle.
Как работает Slope Calculator
Slope is rise over run - vertical change divided by horizontal change. Once you have slope m and any point, the y-intercept comes from y = mx + b ⇒ b = y − mx. The angle with the x-axis is arctan(m).
Formula
m = (y₂ − y₁) / (x₂ − x₁) y = m·x + b θ = arctan(m)
- mSlope (gradient)
- bY-intercept
- θAngle with positive x-axis
Как Slope Calculator получает результат
Slope formula: m = (y₂ − y₁) / (x₂ − x₁), often expressed as 'rise over run.' The order of points doesn't matter as long as you're consistent - switching both numerator and denominator order yields the same result. Vertical lines (x₂ = x₁) have undefined slope; horizontal lines (y₂ = y₁) have slope 0.
From slope to line equation: once you have slope m and any point (x₁, y₁), the y-intercept b = y₁ − m × x₁. So the full line equation in slope-intercept form is y = mx + b. Alternative point-slope form: y − y₁ = m(x − x₁). Both describe the same line; pick whichever is more convenient.
Angle with x-axis: θ = arctan(m), measured anticlockwise from positive x-axis. Slope 1 corresponds to 45°; slope √3 to 60°; slope 0 to 0° (horizontal); slope -1 to 135° (or -45°). Practical interpretations: gradient as percentage = m × 100% (so slope 0.08 = 8% gradient, common road sign value); ratio form like 1:12 means slope 1/12 ≈ 0.083 or about 4.76°.
Разобранные примеры
Roof pitch: 4:12 (4 inch rise per 12 inch run)
Slope = 4/12 = 0.333. Angle = arctan(0.333) ≈ 18.4°. Common Indian residential roof pitch for monsoon water drainage. Steep enough to shed water; gentle enough for easy construction.
Wheelchair ramp accessibility: 1:12 standard
Slope = 1/12 = 0.0833. Angle = arctan(0.0833) ≈ 4.76°. Maximum allowed slope for accessible ramps under most building codes including Indian Harmonised Guidelines.
Когда использовать: Slope Calculator
- Class 9-12 coordinate geometry - slope of a line through two points.
- JEE Main / Advanced - coordinate geometry, motion graph problems.
- Physics - velocity from position-time graph, acceleration from velocity-time graph.
- Civil engineering - road grade, drainage slope, roof pitch.
- Accessibility design - wheelchair ramps must be ≤ 1:12 slope.
- Surveying and topographic analysis - terrain steepness.
- Statistics - regression line slope (least squares).
- Stock and chart analysis - trendline slope as momentum indicator.
Типичные ошибки и подводные камни
- Forgetting that vertical lines have undefined (not infinite) slope.
- Mixing up the order of subtraction - must use (y₂ - y₁) / (x₂ - x₁), not (x₂ - x₁) / (y₂ - y₁) which gives the reciprocal.
- Confusing slope with angle. Slope is dimensionless ratio; angle is degrees or radians.
- Reading roof pitch backwards - 12:4 vs 4:12. Indian convention is rise:run = 4:12 for ~18° pitch.
- Forgetting that parallel lines have the same slope; perpendicular lines have slopes that are negative reciprocals of each other (m₁ × m₂ = -1).
Как получить больше от инструмента
- Convert slope to percentage: m × 100% (useful for road grade signs and ramp specifications).
- Convert slope to angle: arctan(m) in degrees (most practical for construction).
- Parallel lines have equal slopes; perpendicular lines have slopes multiplying to -1.
- Slope of horizontal line is 0; slope of vertical line is undefined.
- Pair with the distance, area and trigonometry calculators on this site for related coordinate geometry work.
Вопросы: Slope Calculator
What if x₁ = x₂?
The line is vertical - slope is undefined (infinite). The line equation becomes x = constant.
Slope of a horizontal line?
Zero. y₂ = y₁, so the numerator is 0 and m = 0. The line is y = b.
Positive vs negative slope?
Positive: line rises left-to-right. Negative: falls. Magnitude tells you steepness - slope of 2 is twice as steep as slope of 1.
How is slope used in real life?
Road grade (% slope), roof pitch, wheelchair ramp gradients (max 1:12 by ADA), staircase rise/run ratios.
Slope-intercept vs point-slope form?
Slope-intercept: y = mx + b. Point-slope: y − y₁ = m(x − x₁). Same line, different form. Calculator gives slope-intercept; converting is one rearrangement.
What's slope's connection to derivatives?
The slope between two points becomes the derivative as those points become infinitesimally close. Calculus generalises slope to curves.
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