Normal Line Calculator
Equation of the normal line to a curve at a point.
Enter your values
Enter the tangent slope at the point you are working with.
Enter the point x1 you are working with.
Enter the point y1 you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Normal line slope
- -0.5
- Normal line equation
- y = -0.5x + 3.5
- Point
- (1, 3)
Quick answer
The normal line is perpendicular to the tangent, so its slope is the negative reciprocal of the tangent slope.
y - y1 = (-1/m)(x - x1)
At a glance
| What it does | Equation of the normal line to a curve at a point. |
|---|---|
| Category | Math |
| Inputs needed | Tangent slope at the point, Point x1, Point y1 |
| Main output | Normal line slope |
| Formula | y - y1 = (-1/m)(x - x1) |
| Cost | Free — no sign-up, no download |
How to use the Normal Line Calculator
- 1Find or enter the tangent slope at the point of interest.
- 2Take the negative reciprocal to get the normal line's slope.
- 3Use point-slope form with the given point to get the full equation.
Inputs explained
- Tangent slope at the point
- Enter the tangent slope at the point you are working with.
- Point x1
- Enter the point x1 you are working with.
- Point y1
- Enter the point y1 you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Tangent slope at the point2
- Point x11
- Point y13
Results
- Normal line slope-0.5
- Normal line equationy = -0.5x + 3.5
- Point(1, 3)
Frequently asked questions
What if the tangent is horizontal?
The normal line is vertical (x = x1) and has no defined slope.
Why is the normal line useful?
It's used in optics, curvature and finding perpendicular distances to curves.
What do I need to enter into the Normal Line Calculator?
Just 3 values: tangent slope at the point, point x1, point y1. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Normal Line Calculator work out the answer?
The normal line is perpendicular to the tangent, so its slope is the negative reciprocal of the tangent slope. It applies the formula y - y1 = (-1/m)(x - x1) and shows the working so you can check each step by hand.
Is the Normal Line Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.